1
The Real Numbers
(2001 – 2nd Session)
Consider the numbers C and D:
𝐶 = 3√12 + √27 ;
𝐷 = (2√3 + 3)
2
1) Write C under the form 𝑎√𝑏 where 𝑎 and 𝑏 are two integers.
2) Write D under the form 𝑝 + 𝑞√3 where 𝑝 and 𝑞 are two integers.
(2002 – 1st session)
Given the two numbers A and B:
𝐴 = 2√27 − 2√3 + √12 ; 𝐵 = √75 + √48 − 7√3
1) Show that A = 𝑎√3 and B = 𝑏√3 where 𝑎 and 𝑏 are two integers to be determined.
2) Calculate 𝐴 × 𝐵.
A
3) Show that the opposite table is a proportionality table
4√5 + 2√11
(2002 – 2nd Session)
Consider the following three numbers:
14 27 3
14 × 105
𝐴 = √12 − 2√27 + 4√75 ; 𝐵 =
×
÷
𝑎𝑛𝑑 𝐶 =
45 49 5
0.7 × 102
1) Write 𝐴 in the form 𝑎√3 where 𝑎 is an integer.
2) Write 𝐵 in the form of an irreducible fraction.
3) Write 𝐶 in the form 𝑏 × 10𝑛 , where b is an integer.
(2003 – 1st Session)
Write each of the following numbers as a fraction in its simplest form:
3 2 1
𝐴=( ) −
7
7
;
5
−1
𝐵=3
1
1−6
;
𝐶=
4 × 1012 × 1.5
9 × 1011
(2003 – 2nd Session)
Work out and give the result in its simplest form, writing all the steps of calculation:
5 9 1
𝐴= − ×
2 2 3
1014 × 210
𝐵=
5 × 4 × 1012 × 29
2
4√5 − 2√11
B
(2004 – 1st session)
In this problem, the unit of length is 1cm.
Given the three points A, B and C such that: AB = √108 , BC = √48 and AC = 10√3
1) Calculate AB + BC giving the answer in the form 𝑎√3.
2) Are the points A, B and C collinear? Justify.
(2004 – 2nd session)
Consider the following numbers A, B and C:
1  2
A=  
5 5
2
B = (2  5 )  2(8  20 ) ,
2
,
C=
 1.25  8  10 7  10  4
4  10 2
.
1) Calculate A, B and C showing all the steps of calculation and give each result in its simplest possible form.
2) From the numbers A, B and C, choose two opposite numbers and two numbers inverses of each other.
(2005 – 2nd session)
2
3
Given that: A =
1
2
3
1
,
B
5  108  2  10 3
7  (10 4 ) 3
,
45  80  2 125
C=
7  35  7 5  3
.
1) Calculate A and B, showing all the steps of calculation, and deduce that A and B are two forms of same number.
2) Calculate C and give the result in the form 𝑎√5 where 𝑎 is an integer.
(2006 – 1st session)
Given the two numbers A and B:
A=
3.6×103 ×10−5
9×102
2
and B = (2 + √5) + √5(1 + 2√5)
1) Write A in the form 𝑎 × 10𝑛 where 𝑎 and 𝑛 are two integers, then write A in the form of a decimal number.
2) Write B in the form 𝑏 + 𝑐 5 where 𝑏 and 𝑐 are two integers.
(2006 – 2nd session)
Write each of the following numbers in the form of a fraction as simple as possible:
A
7 8 5
 
3 3 2
;
5
1
3
B
;
1
1
6
C
3
8  10 7  1.5
3  109
.
(2006 – 2nd session)
Given the two numbers: X  32  3 2  2 18
;
Y  50  72  3 2 .
1) Write X in the form 𝑎√2 and Y in the form 𝑏√2 where 𝑎 and 𝑏 are two integers to be calculated.
2) Deduce that X  Y = 28.
X
4√3 + 2√5
3) Prove that the opposite table is a proportionality table.
Y
4√3 − 2√5
(2007 – 1st session)
Given the two numbers A and B defined by:
13 3 14
A=
 
, B  2 36  5 12  9 75  4 27 .
7
7 9
1) Calculate A and give the result in the form of an irreducible fraction.
2) Write B in the form 𝑎 + 𝑏
3 where 𝑎 and 𝑏 are two integers.
(2007 – 2nd session)
Given that: A 
8 3 14
 
and
7 7 6
B
2.1  104  105
.
3  102
1) Write A in the form of an irreducible fraction.
2) Write B in the form 𝑎 × 10𝑛 where 𝑎 and 𝑛 are two integers.
(2008 – 1st session)
Given the two numbers A and B:
2
𝐴 = (√3 + √2) + (√3 − √2)(√3 + √2) ; 𝐵 = √50 + √150 + √96 + √54 − 5√2
1) Calculate A and write it in the form of 𝑎 + 𝑏√6 where 𝑎 and 𝑏 are integers.
2) Calculate B and write it in the form of 𝑥√6 where 𝑥 is an integer.
3) By using the answers in questions 1 and 2, rationalize the denominator of the expression
A
and
B
simplify the answer.
(2008 – 2nd session)
The questions 1), 2) and 3) are independent of each other.
1) Write the following expression in the form of a decimal fraction, showing the steps of your
calculation:
2.4  52  3(9.3  4.3)2
2  2.5  60
.
4
2) Given that a =
3
2
,b =
3
4
and c =
4
5
. Calculate (b – a), (a – bc) and
ac
b
.
3) x is a natural integer:
a. Write in terms of x the natural integer which is just before x and the integer that is just after x.
b. Calculate the product of these three natural integers in terms of x.
c. Use the above result to calculate: 9  10  11.
(2009 – 1st session)
Given: A = 2√27 + 3√75 − 3√48 𝑎𝑛𝑑 B =
22
√18−√8
1) Write A in the form 𝑎√3 and B in the form 𝑏√2 where 𝑎 and 𝑏 are two integers.
2) Compare A and B and justify.
3) Show that A − B =
1
A+B
(2009 – 2nd session)
Consider four distinct points O, K, L and S such that:
OK =
2×103 [(2×10−1 )2 +(6×10−2 )]
5×10−1 ×2×102
5
2
6
15
; OL = 20 ( −
3
(√10−√2)(√10+√2)
5
√16
− ) and OS =
1) Calculate, in showing the details of calculation OK, OL and OS and write each result in the form of a
natural number.
2) Show that the points K, L and S belong to the same circle with center O.
(2010 – 1st session)
Consider the three numbers A, B and C.
3 1
5
3 × 103 × 1.2 × 10−5
A = − × ( + 2) , B =
and C = √63 − 2√28 + √700
5 5
2
15 × 103
1) Calculate A and write the answer as a fraction in the simplest form, then as a decimal.
2) Write B in scientific notation.
3) Write C in the form 𝑎√𝑏 where 𝑎 and 𝑏 are two integers, then give to the nearest thousandth an
approximate value of C.
(2011 – 1st session)
Given: A =
8
3
5
21
3
15
− ×
; B=
3.4×10−3 ×5×(102 )3
4×10−3
𝑎𝑛𝑑 C =
(1−√3)
(2+√3)
1) Calculate A and write the answer as a fraction in the simplest form.
2) Calculate B and write the answer in scientific notation.
3) Write C in the form 𝑎 − 𝑏√3 where 𝑎 and 𝑏 are two integers.
5
2
2
(2011 – 2nd session)
1) Calculate the greatest common divisor (GCD) of 154 and 112.
2) Write the fraction
3) Let m =
154
112
+
1
154
112
in its simplest form.
8
a. Write m as a fraction in the simplest form.
b. Is the number m decimal? Justify.
(2012 – 1st session)
Consider the three numbers A , B and C :


4
10 3
3 5 7
A  5  1  5 1 ; B   
and C 
.
4 4 15
25 103  3  103  6  37.5
In the following questions, the steps of calculation must be shown
1) Prove that A is a natural number.
2) Write B as a fraction in its simplest form.
3) Prove that C is a decimal number.

 
2

2
(2012 – 2nd session)
Consider the following numbers A, B and C:
2 2 1
5 2
3
18 × 108
𝐴=( ) +
; 𝐵 = − ÷ (1 + ) ; 𝐶 =
3
3
3 3
2
8 × 107 × 3.5
1) Write, showing all the steps of calculation, each of the numbers A, B and C as a fraction in its
simplest form.
2) Out of the found fractions, indicate that which is decimal. Justify.
(2013 – 1st session)
Given: A =
5√18−2√98
2√3×√24−4√2
1) Write A as a fraction in its simplest form.
2) Write A in scientific notation.
(2013 – 2nd session)
All steps of calculation must be shown in each exercise.
1) Given A = 6 × 102 + 102 + 4 × 10−2 + 10 − 6
a. Write A in the form of a decimal number.
b. Write A in scientific notation.
c. Write A as the sum of an integer and a fraction less than 1, in its simplest form.
2) Show that the number D =
4
2+√3
÷
2−√3
is a natural number.
2
6
3) Given the two numbers B =
5
8
3
4
3
8
6
2
2
+ × − ( − 1) 𝑎𝑛𝑑 C = 2√75 − 4√27 + 4√12
a. Write B as a fraction in its simplest form.
b. Write C in the form 𝑎√3 where 𝑎 is an integer.
(2014 – 1st session)
1) Rationalize the denominators of the following fractions
2) Consider a triangle ABC such that AB =
6
√7−1
6
√7−1
6
; AC =
𝑎𝑛𝑑
√7+1
6
√7+1
and BC = 4.
a. Calculate AB2 and AC2 . Deduce that the triangle ABC is right at A.
b. Let M be the midpoint of [BC] and E a point on the ray (semi-line) [AM) such that AE =
Calculate AM and ME.
(2015 – 1st session)
Consider the three numbers A, B and C so that:
A=
8
2
6
6
2√75 − √48
+ 5 ÷ (1 − ) ; B = √2 − × √2 +
𝑎𝑛𝑑 C =
3
5
5
5
5√2 × √54 − 5√27
In what follows, the steps of calculation must be shown.
1) Show that A is a natural number.
2) Write B in the form of a fraction in its simplest form.
3) Prove that C is decimal
4) Prove that BC2.
(2015 – 2nd session)
Given the number a =
7+√125+√20
14
1) Write a in the form 𝑥 + 𝑦√5 where 𝑥 and 𝑦 are two natural numbers.
2) Compare 𝑎 + 1 and 𝑎2 .
3) Verify that 𝑎3 = 2𝑎 + 1.
(2016 – 1st session)
Consider the three numbers A, B and C:
9 9 1
A  
2 2 3
1014  210
; B
5  4  1012  29

 
2

2
; C  2  5  1 2 5 .
1) By writing all the steps of calculation, show that A, B and C are natural numbers.
2) Verify that A  B  C .
7
10
3
.
(2016 – 2nd session)
Consider the three numbers A, B and C:
1 7 5
5 × 10−2 × 7 × 105
A= + ÷
; B=
𝑎𝑛𝑑 C = √45 − 4√5 + 3√125
3 6 3
2 × 107
1) Calculate A and write the answer as a fraction in the simplest form.
2) Calculate B and write the answer in scientific notation.
3) Write C in the form 𝑎√5 where 𝑎 is a natural number.
(2017 – 1st session)
The questions 1) and 2) are independent. All the steps of calculation must be shown.
1
1) Given A =
2−3
1 and B =
2+3
24×103 ×5×106
8×(103 )3
a. Calculate A and give the result as a fraction in its simplest form.
b. Show that B is a natural number.
2) Given C =
2
√45−√180+9
and D = (1 − √5)
3+√5×√35−5√7
a. Write C in the form 𝑛 − √5 where 𝑛 is a natural number.
b. Calculate D, then verify that D = 2 × C
(2019 – 1st session)
Consider the three numbers A, B and C so that:
𝐴 = √18 − √8 + √50 , 𝐵 =
1
√2+1
2
𝑎𝑛𝑑 𝐶 = (√2 + 1) + 1
1) Write A in the form m 2 where m is an integer.
2) Show that B = 2  1 .
3) Write C in the form n 2  p where n and p are integer.
4) Show that B × A × C is an integer.
(2019 – 2nd session)
Given A = 80  20  5 .
1) Write A in the form m 5 where m is an integer.
2) Let B = 5 5 .
a. Show that the adjacent table is a proportionality table.
20
b. Write
in the form p  5 where p is an integer.
B5
8
A
2 19  1
2 19  1
B
(2019 – 3rd session)
Answer by True or False. Justify your answer.
1
 52 6 .
x
1) If x = 5  2 6 , then
109  108
2) If N =
, then N = 109  1 .
8
10
3) If x + 1 ≤ 3 , then x ≥ 3  1.
***Multiple choice questions:
No//
Questions
Answers
b
38
45
26
25
c
22
15
1
3
2
4 3 10
− ×
=
5 5 6
a
2
3
1
−
5
3
314 − 312 =
32
312 × 8
6
4
If 215 − 𝑥 = 214 , then 𝑥 =
214
2
27
5
If (x + x) = 15 then 𝑥 2 + 𝑥 2 =
225
13
√15
8 × 101
12𝑛+1
8 × 10−1
10√2
0
−30√2
1
8
15
6
+
7
2
15
× is equal to…
3
1 2
1
4 𝑛+1
𝑛 is a natural number (5)
7
3√2 − √50 + √8 =
8
1
√5 − 2
5 𝑛
× (4) =
=
√5 + 2
√5 + 2
3
1
21
-1
6
7
9
1 1 6
− × =
3 3 7
10
(3 + √5) − 14 =
9 + √5
0
6√5
11
If (√2 − 1)x = 1 then x =
√2
1
√2 + 1
12
n is a non-zero real number, 2 − 2 × 3 =
3
-n
0
0
2
n
n
9
Algebraic expressions
(2001 – 1st Session)
Given P(x) = 2(x − 1)(x − 2) + x 2 − 4
a. Factorize P(x).
b. Solve the equation P(x) = 0.
(2001 –2nd Session)
Given the algebraic expression E = (2x + 1)(4x − 6) − (2x − 3)2
1) Prove that E can be written under the developed form: E = 4x 2 + 4x − 15.
3
2) Calculate the values of E for x = 2 and for x = √2.
3) Factorize E then solve the equation E = 0.
(2002 – 1st Session)
Given the expression E(x) = (x – 2)(3x + 1) – (x – 2)(x + 10).
1) Factorize E(x).
2) Solve the equation E(x) = 0.
3) In this question, x is a length expressed in cm such that x > 2.
ABCD is a rectangle such that AB = x – 2 and BC = 3x + 1.
MNP is a triangle, right angled at M, such that MN = 2(x – 2) and MP = x + 10.
Designate by S the area of ABCD and S′ the area of MNP.
a. Express S and S′ in terms of x and show that S – S′ = E(x).
b. Calculate x so that S = S′.
(2002 –2nd Session)
Given that E = (2x – 3)2 + (x + 1)(5x + 7).
1) Expand (2x – 3)2.
2) a. Show that E = 9x2 + 16.
b. Find the values of x for which E = 52.
(2003 – 1st Session)
Given the expression E(x) = (3 – 5x)2 + (25x 2 – 9) + 2(3 – 5x) (– x + 2).
1) Expand and reduce E(x).
2) Factorize 25𝑥 2 – 9, then factorize E(x).
3) Solve the equation E(x) = 0.
(2003 –2nd Session)
1) Expand and reduce E(x) = (2x + 1) (2x – 1).
2) Calculate E(x) for x = √7 .
3) Explain how we can use the first question to calculate the product 201×199, then calculate this
product.
10
(2004 – 1st Session)
Consider the two expressions: A(x) = (x + 3)(4x + 7) and B(x) = x 2 – 4 + (x – 2)(3x + 5).
1) Solve the equation A(x) = 0.
2) Prove that B(x) = (x – 2) (4x + 7).
x2 −4+(x−2)(3x+5)
3) Given the expression F(x) =
(x+3)(4x+7)
a. Determine the values of x for which F(x) is defined.
b. Simplify F(x); then solve the equation F(x) = 2.
c. Does the equation F(x) = – 3 admit a solution? Justify.
(2004 –2nd Session)
1) Determine the numerical values of 𝑎 and 𝑏 of the polynomial P(x) = ax 2 + bx + 2a – 3b – 9 so
that P(1) = 0 and P(2) = 0.
2) Given the polynomial Q(x) = (x – 1)(x – 2).
a. Show that Q(x) – 2 = x (x – 3).
b. Solve the equation Q(x) = 2.
(2005 – 1st Session)
Given the expression E = (2x + 3)2 + (x – 1)(2x + 3).
1) Expand and reduce E.
2) Calculate the exact value of E for x = 2 .
3) Factorize E.
4) Solve the equation: (3x + 2)(2x + 3) = 0.
G
S3
S4
S1 S1
D
E
S2
S
C
Q
2x + 1
2x +1
(2006 – 1st Session)
1) Consider the expression: E(x) = 4x 2 – 1 + (2x + 1)2 + x (2x + 1).
Show that E(x) = 5x (2x + 1).
2) In the figure below:
A
B
2x + 1
P
H
R
5x
x
F
 𝑥 is a measure of length in centimeters and 2x – 1 0.
 ABCD is a square of area S1 .
 DCFE , HADG and PQRS are three rectangles of areas S2 , S3 and S4 respectively.
a. Express S1 and S2 in terms of x .
b. Knowing that S1 + S2 + S3 = S4 and using the preceding results, calculate AH in terms of x .
11
(2007 – 1st Session)
Part A
1) Verify the equality: 2(x – 3)(x + 7) = 2x 2 + 8x – 42 .
2) Solve the equation :2x 2 + 8x – 42 = 0.
Part B
In this part, the unit of length is the centimeter.
ABC is a triangle such that AB = x, AC = x + 4 and BC = 58 , where x is an integer strictly greater
than 1.
1) Can we find a value for x such that triangle ABC is right angled at C? Justify.
2) Calculate x so that triangle ABC is right angled at A. (You can use the results of part A).
3) Calculate x so that the perimeter of triangle ABC is less than or equal 18.
(In this question, you can consider 7.6 as an approximate value of
58 ).
(2007 –2nd Session)
Given that P(x) = 4x 2 − 9 + (x − 2)(2x + 3) and Q(x) = (2x + 3)(x − 1)
1) Prove that P(x) = (2x + 3)(3x − 5)
2) Solve the equation Q(x)  0 .
P(x)
.
Q(x)
a. For what values of x, is F(x) defined?
3) Let F(x) 
b. Simplify F(x), then solve the equation F( x)  2 , and write the solution in the form
where a, b and c are integers.
(2008 – 1st Session)
Consider the polynomial P(x) = (x – 2)2  (2 – x)(x + 4)
1) Factorize P(x).
2) Expand and reduce P(x).
3) a. Expand and reduce 2(𝑥 – 3)(𝑥 + 2).
b. Calculate P(3).
c. Solve the equation P(x) = 8.
(2008 –2nd Session)
1) Consider the polynomial 𝑃(𝑥) = 𝑎 𝑥 2 – 4(𝑥 + 5).
a. Calculate 𝑎 such that P(−2) = 0.
b. Let E(x) = 4(x2 – 4) – (x + 2) 2 ; verify that E(x) = 3 x2 – 4x – 20.
c. Factorize E(x).
d. Solve the equation E(x) = 0.
12
ab 2
c
2) Suppose that x = 2 2  1
a. Calculate 𝑥 2 and 2𝑥 + 7, and then compare the two numbers obtained.
b. Verify that x – 2 =
7
.
x
(2009 – 1st Session)
Consider the polynomial P(x) = (x + 9)2 − 3(x − 1) (x + 9).
1) Factorize P(x).
2) In this part, the unit of length is the centimeter.
In the figure to the right, ABCD is a square, DEC is a triangle such that:
CF = 9, DF = x and the height EF = x – 1 with x > 1.
Calculate x such that the area of the square ABCD is 6 times the area of the
triangle CED.
(2009 –2nd Session)
Given E(x) = (2x – 1)2 + (x – 2)(1 – 2x) and F(x) = ax 2 + bx – 2.
1) Factorize E(x).
2) Calculate 𝑎 and 𝑏 such that F(1) = 5 and F(– 2) = 20.
3) Let Q(x) = 6x 2 + x – 2. Verify that Q(x) = (2x – 1)(3x + 2).
4) Let 𝑃 (𝑥) =
𝐸(𝑥)
𝑄(𝑥)
a. Determine the values of x so that P(x) is defined. Then simplify P(x).
b. Solve the equation P(x) = 0.
3
c. Does the equation P(x) = have a solution? Justify.
7
(2010 – 1st Session)
Given A(x) = 2(2x – 3)(x – 4) + (8x 2 – 18) – 2(2x – 3)2
1) Show that 8x 2 – 18 = 2(2x – 3)(2x + 3).
2) Factorize A(x).
3) Solve the equation A(x) = 0.
4) Let B(x) = 2x 2 + 8x + 8. Factorize B(x).
5) Let F(x) =
A(x)
B(x)
.
a. Find x so that F(x) is defined.
b. Simplify F(x), then solve the equation F(x) = 3.
(2010 –2nd Session)
Given: P(x) = – x 2 + 6x – 8 and Q(x) = (x – 2)2 – 3(x – 2).
1) Show that P(x) = (x – 2)(4 – x), then solve the equation P(x) = 0.
2) Factorize Q(x).
3) Given F(x) =
𝑄(𝑥)
𝑃(𝑥)
a. For what values of x the expression F(x) is defined?
13
b. Simplify F(x), then solve the equation F(x) = 1.
3
c. Does the equation F(x) = − admit a solution? Why?
2
(2011 – 2nd Session)
The two parts A and B of this exercise are independent.
Part (A):
Given P(x) = (3x – 2)(x + 2) – (3x – 2)2
1) a. Develop and reduce P(x).
b. Calculate P(√5).
2) a. Factorize P(x).
b. Solve the equation P(x) = 0.
Part (B):
Given the two real numbers x and y such that 𝑥𝑦 = 2√3 and +𝑦 = 2 + 2√3 .
1) Calculate x2y + xy2. Give the result in the form 𝑎 + 𝑏√3 where 𝑎 and 𝑏 are two integers.
2) Calculate x2 + y2.
(2012 –1st Session)
Consider the expressions: A(x) = 4x 2 – 9 and B(x) = (2x – 3)2 – (2x – 3)(x – 5)
1) Factorize A(x).
2) a. Verify that B(x) = (2x – 3)(x + 2).
b. Solve the equation B(x) = 0.
3) Consider the expression F(x) =
A(x)
B(x)
a. For what values of x, F(x) is defined?
b. Simplify F(x) and solve F(x) = 3.
c. Calculate F(√5) and write the answer in the form 𝑎 − 𝑏√5 . (a and b are natural numbers)
(2012 –2nd Session)
Consider the following expressions: E = (x + 9)2 – 25 and G = (x + 4)(x + 14) – 2(x + 4)2 .
1) Verify that E = (x + 4) (x + 14) and factorize G.
2) The adjoining diagram is that of an apartment in the form of a
square with side (x + 9) meters (x > 0).
It is formed of a salon, a room and a kitchen.
The room is a square with side 5 meters and the kitchen is also a
square with side (x + 4) meters.
a. Express, in terms of x, the area A of the apartment and
calculate the area A1 of the room.
b. Determine A2, the sum of areas of the salon and the kitchen.
c. Express, in terms of x, the area A3 of the kitchen.
Determine x so that A2 is the double of A3.
14
(2013 –1st Session)
Given that A(x) = (2x – 3)2 – (x – 6)2 and B(x) = 2(x – 3)2 + 9 – x 2 .
1) a. Expand and reduce A(x).
b. Calculate A(1 + √2) ,Write the answer in the form 𝑎 + 𝑏√2 where 𝑎 and 𝑏 are two integers.
2) Factorize A(x).
3) Verify that B(x) = (x – 3)(x – 9).
4) Let F(x) =
A(x)
B(x)
a. For what values of x, ix F(x) defined?
b. Simplify F(x), then solve the equation F(x) = – 1.
(2013 –2nd Session)
Part A:
Given E (x) (3x 1)2 (3x 1)(x 2)
1) Expand and reduce E(x).
2) Calculate x such that E(x) = 3.
3) Factorize E(x).
Part B:
In the next figure:
 x represents a length in cm so that x > 0.5
 ABCD is a square with side 3𝑥 − 1.
 DEFG is a rectangle such that:
FG = x + 2 and EF = 3x – 1
1) Calculate, in terms of x, the area S of ABCD and the area S' of DEFG.
2) Solve the equation S-S' = 0.
3) Determine all the integers 𝑥 so that: S S ' 6x2 5x 12.
(2014 –1st Session)
Given the following polynomial: E(x) = (x – 3)2 + (x + 7)(x – 3).
1) Develop and reduce E(x).
2) Factorize E(x).
3) ABCD is rectangle with AB = 2x and BC = x – 1.
Calculate in terms of x the area S of the rectangle. Determine x such that S = 12.
(2015 –2nd Session)
1) a. Verify that x 2 + 4x + 3 = (x + 2)2 – 1
b. Factorize x 2 + 4x + 3.
2) Given an isosceles triangle ABC with vertex A so that its area is equal to x 2 + 4x + 3 and BC =
2x + 2 (x > 0). Let [AH] be an altitude in this triangle.
a. Show that AH = x + 3.
b. Calculate AB2 in terms of x.
3) Find x such that the area of ABC is equal to 8.
15
(2015 –1st Session)
Given the algebraic expression:
E(x) = (3x – 4)2 – (3x – 4)(x + 2).
1) a. Show that E(x) = 6x 2 – 26x + 24.
b. Solve the equation E(x) = 24.
2) Factorize E(x).
3) In the adjacent figure:
ABCD is a square with side 3𝑥 – 4.
AMND is a rectangle such that DN = x + 2.
(x is real number greater than 3).
a. Express, in terms of x, the area S of the square ABCD
and S’ the area of the rectangle MBCN.
b. Determine x so that S = S’.
(2016 –1st Session)
Given the expression: E(x) = (3x + 1)(2x − 1) − (3x + 1)(x + 1).
1) Show that E(x) = (3x + 1)(x − 2)
2) Solve the equation E(x) = 0.
3) In the adjacent figure:
 x is a length expressed in cm such that x > 1.
 ABCD is a rectangle such that AB  3x  1 and BC  2x 1 .
 ABM is a triangle, right angled at B, such that MB  2(x  1) .
Denote by S the area of ABCD and S' that of ABM.
a. Express S and S' in terms of x.
b. Verify that S – S′ = E(x).
c. Calculate x so that S = S′.
(2016 –2nd Session)
Given:E(x) = 5(x − 1)(x + 2) − (x + 2)2 + 3(x + 5).
1) Show that E(x) = 4x 2 + 4x + 1
2) Solve the equation E(x) = 1.
3) Consider H(x) = 9x 2 − (2x + 1)2 .
a. Show that H(x) = (5x + 1)(x − 1).
b. Solve the equation H(x) = 0.
16
(2017 –1st Session)
Given A(x) = (2x − 3)2 + (x − 5)(3 − 2x)
1) Factorize A(x)
2) Let B(x) = 2x 2 − 5x + 3. Verify that B(x) = (2x − 3)(x − 1).
3) Let F(x) =
(2x−3)(x+2)
B(x)
a. For what values of x, is F(x) defined?
b. Simplify F(x).
c. Does the equation F(x) = 7 have a solution? Justify.
(2017 –2nd Session)
Given A(x) = (x − 3)2 − (x − 3)(2x − 7).
1) Prove that A(x) = (x − 3)(4 − x).
2) Let B(x) = (16 − x 2 ) + A(x). Factorize B(x).
3) Let F(x) =
A(x)
(4−x)(2x+1)
.
a. For what values of x, is F(x) defined?
b. Simplify F(x), then solve the equation F(x) =
2
3
c. Does the equation F(x) = x have a solution? Justify.
(2018 –1st Session)
Given A(x) = (3x − 2)2 − (2x − 1)(3x − 2) and B(x) = 9x 2 − 4
1) a. Verify that A(x) 3x 2x 1.
b. Solve the equation A(x) 0.
2) Factorize B(x).
3) Let F(x) =
(3x−2)(3x+2)
A(x)
a. For what values of x, is F(x) defined?
b. Simplify F(x).
c. Does the equation F(x) 12 admit a solution? Justify.
(2019 –1st Session)
1) Given P(x) =  2x  1  2x 2  9x  4 .
2
a. Verify that  2x  1 x  4   2x 2  9x  4 .
b. Show that P(x) =  2x  1 x  3 .
c. Solve the equation P(x) = 0.
17
Px
.
4x 2  1
a. Factorize 4x 2  1 .
b. For what values of x, is H(x) defined?
c. Simplify H(x).
2) Let H(x) =
(2019 –2nd Session)
In the adjacent figure:
ABC is a right angled triangle at A; AB = 6 and AC = 8.
B
M is a point of [AB] and N a point of [AC] so that:
x
AN = BM = x (0 < x < 6).
M
Denote by S the area of triangle ABC and S′ that of triangle AMN.
1) Calculate S.
2) Calculate AM in terms of x and show that S′ =
6x  x 2
.
2
A
3) a- Verify that: 3  x  2  x  4   3x 2  18x  24 .
b- Calculate x so that S = 6S′.
9 1
2
4) a- Show that S′ −   x  3 .
2 2
b- Deduce that the area of triangle AMN is less than or equal to
(2019 –3rd Session)
Given A  x    x  2    4  2x  x  3 .
2
1) Show that A  x    x  2  3x  4  .
2) Solve the equation A  x   8 .
3) Let H  x  
2  x
.
Ax
2
a. For what values of x, is H(x) defined?
b. Simplify H(x).
c. Does the equation H(x) = 0 admit a solution? Justify.
18
9
.
2
x
N
C
Inequalities
(2001 – 2nd Session)
Given the inequality: x + 5 < 4 (x + 1) + 7.
1) Study if each of the following numbers is a solution of this inequality:– 5; 0 ; – 2.
2) Solve this inequality and represent the solution on an axis of origin O.
(2002 – 2nd Session)
3
5
2
2
Solve the following inequality: x − ≤ − x + 1
(2004 – 2nd Session)
Given the inequality: 2x + 1 < 5(x – 1) + 15.
Solve this inequality and represent the solutions on an axis of origin O.
(2005 – 1st Session)
Solve the following inequality: 4(2x – 1) ≥ 9x – 7
(2005 – 2nd Session)
Let E =
4x+7
3
1) Calculate the value of E for x =
7
4
4x+7
2) Without solving the inequality 3 < 5, is the number 7 a solution of this inequality? Justify.
(2007 – 2nd Session)
A video-club offers its customers two choices A and B. Each choice is formed of a fixed sum paid in
advance which is called subscription and another sum to be paid for each rented cassette (Film).
Choice A
Choice B
Subscription in LL
60 000
42 000
Price in LL paid for each rented cassette
900
1 500
1) A customer who wants to rent 20 cassettes chooses choice A. How much should he pay?
2) Designate by x the number of cassettes that a second customer desires to rent.
a. Express in terms of x, the price S1(x) that this customer should pay if he chooses choice A, and the
price S2(x) that he should pay if he chooses choice B.
b. Starting from which number does the rented cassettes of choice A become more advantageous
than choice B? (We advise you to start by solving the inequality S1(x) ≤ S2(x))
3) A third customer has chosen choice B and paid 93 000 LL.
a. What is the number of cassettes rented to this customer?
b. Which choice is better for him? Justify.
19
(2010 – 1st Session)
In this exercise, the unit of length is the centimeter.
̂ C = 900
In the adjoining figure, BA
AB = 4, AM = 1, AC = x (x > 0) and AMNE is a square.
1) Express in terms of x the area of the triangle ABC.
2) Given the information: the sum of areas of the triangle ABC
and the square AMNE is greater than 11.
a. Write an inequality modeling the previous information.
b. Solve this inequality, then compare AB and AC.
(2011 – 1st Session)
An agency for renting cars proposes to its customers the following two offers A and B:
Deposit
Charge per Km
Offer A
50 000 LL
600 LL
Offer B
42 000 LL
700 LL
Denote by x the number of kilometers traveled by a car that a customer rents.
1) Find in terms of x, the amount S paid by this customer if he selects the offer A, and the amount S' if he
selects the offer B.
2) Calculate x so that S is equal to S'.
3) Starting what traveled distance is the offer A more advantageous than the offer B? Justify.
4) A second customer selects offer A and pays 410 000 LL. What is the traveled distance by this
customer?
(2012 – 2nd Session)
The owner of a bookshop proposes the following offer to his clients:
"The first five CD are rented at the rate of 600 LL each, and the others are rented at the rate of 500 LL
each".
A client has rented x CD and paid a sum less than 9 000 LL (x > 5).
1) Show that the previous information are modeled by the following inequality: 500x + 500 < 9 000.
2) Solve this inequality and find the greatest value of x.
(2016 – 2nd Session)
Solve the following inequality and represent the solution on a number line:
3 5
4x − ≤ x + 3
2 2
***Multiple choice questions:
No//
Questions
1
The natural numbers, solutions of 3x − 1 < 8 are…
2
If 2x − 3 > 5, then…
20
a
1;2;3
Answers
b
0;1;2;3
c
0;1;2
x+4 >0
−3x + 12 < 0
x < −4
Proportionality
(2002 – 1st Session)
Given the two numbers A and B:
A = 2√27 − 2√3 + √12 ; B = √75 + √48 − 7√3
1) Show that A = 𝑎√3 and B = 𝑏√3 where 𝑎 and 𝑏 are two integers to be determined.
2) Calculate A × B.
3) Show that the opposite table is a proportionality table
A
4√5 + 2√11
4√5 − 2√11
B
(2002 – 2nd Session)
A box contains a certain number of balls, distributed as follows:
 29% of the balls are red.
 45% are white.
 104 balls are green.
Calculate the total number of balls in this box.
(2003 – 1st Session)
1) Calculate the numbers x and y so that the following table is a proportionality table.
80
2
75
x
5
y
2) To prepare a cake, a pastry-cook uses the following proportions:
Flour: 80 g; Eggs: 2; Sugar: 75 g.
Keeping the same proportions, calculate the quantities of flour and sugar needed to mix with 10 eggs.
(2005 – 1st Session)
The students of a school are distributed in the following way:
 47% are in the elementary section
 27% are in the intermediate section
 130 students are in the secondary section.
1) What is the percentage of students in the secondary section?
2) Calculate the number of students of this school.
(2007 – 2nd Session)
A bag contains a number of balls, distributed in the following way:
 10% of the balls are red.
 15% of the balls are white.

2
5
of the balls are green.
 42 balls are black.
1) Find the percentage of the green balls and that of the black balls.
2) Calculate the total number of balls in the bag.
21
(2009 – 1st Session)
1) A class contains 30 students where 40% of them are boys. Another class contains 20 students where
60% of them are boys. The students of these two classes meet together in the computer room.
Calculate the number and the percentage of the boys in this room.
2) All the articles of a certain shop are subject to an increase of 20% of their prices.
Denote by x the original price of an article and by y its new price after the increase.
a) Find y as a function of x.
b) If the new price of a calculator is 30 000 LL, what is its original price?
(2012 – 2nd Session)
An article costs 18 000 LL. If its price is subject to a discount of 12%, followed by a raise of 15%, what is
therefore the new price of this object?
(2013 – 1st Session)
Determine the real number x so that the following table represents a proportion:
x
3 + √5
7
5
3 − √5
(2016 – 2nd Session)
A box contains 400 balls, distributed as follows:
 30% of these balls are red
 108 balls are green
 The remaining balls are white.
1) Find the percentage of green balls.
2) Calculate the number of white balls.
(2019 – 2nd Session)
Given A = 80  20  5 .
3) Write A in the form m 5 where m is an integer.
4) Let B = 5 5 .
c. Show that the adjacent table is a proportionality table.
20
d. Write
in the form p  5 where p is an integer.
B5
22
A
2 19  1
2 19  1
B
***Multiple choice questions:
No//
Questions
1
If each year prices increase by 10% then at the end
of two years prices will increase by…
2
An object costs 270LL. Its price is increased by
5%. The new price of the object is…
3
a
Answers
b
c
100%
21%
20%
275 LL
270.05 LL
283.5 LL
1 458 LL
1 242 LL
1 250 LL
10%
18%
8%
5 980 L.L.
780 L.L.
4 420 L.L.
17 000 L.L.
20 000 L.L.
19 550 L.L.
4
1
2
√2
2
The price of an article increases by 8% and
becomes 1 350 LL.
The original price of this article is…
4
The original price of an article is 30 000 LL, and
its price after the discount is 27 600 LL. The
percentage of discount is…
5
The original price of an article is 5 200 L.L. After
a discount of 15%, the new price will be…
6
After an increase of 15%, the price of an article
becomes 23 000 L.L.
The original price of this article is:
7
x
√2
√2
4
The table above is a table of proportionality for
x=
8
A car costs 15 000 000 LL.
1 650 000 LL 13 350 000 LL
After a reduction of 11%, its price becomes
23
16 650 000 LL
System of equations
(2001 – 1st Session)
To buy 5 copybooks and 4 pens we must pay 7950 LL.
To buy 2 copybooks and 4 pens we must pay 5700 LL.
1) Translate the given information by s system of two equations with two unknowns.
2) Calculate the price of a copybook and the price of a pen.
(2001 – 2nd Session)
To buy two copybooks and three pens, we must pay 9000 LL. To buy four copybooks and two pens, we
must pay 8000 LL.
2x + 3y = 9000
The preceding givens are translated by the following system: {
4x + 2y = 8000
1) What do x and y represent in this system?
2) What is the information translated by the equation 2x + 3y = 9000?
3) Calculate the price of a copybook and the price of a pen.
(2002 – 1st Session)
a
3
If we add 5 to each term to the fraction , we get a fraction equal to . If we subtract 10 from each term
b
4
a
3
of the fraction , we get fraction equal to . Calculate a and b.
b
5
(2002 – 2nd Session)
A cake-shop sells, on the same day, 15 pies. Some of these pies are apple-pies and the others are
strawberry-pies. The price of one apple-pie is 150 LL while the price of one strawberry pie is 200 LL. The
total selling price of all pies is 2600 LL.
1) Translate the given information into a system of two equations with two unknowns.
2) Solve the obtained system, showing the followed steps in details, and find the number of apple-pies
and that of strawberry-pies.
(2003 – 2nd Session)
A flower shop sells Tulips and Roses. A bunch of 2 Tulips and 4 Roses costs 13 000 LL. A bunch of one
Tulip and 5 Roses costs 14 000 LL.
1) Find by writing all the followed steps, the price of one Tulip and the price of one Rose.
2) Form a bunch of 6 flowers which costs 10 000 LL.
(2004 – 2nd Session)
Determine the numerical values of a and b of the polynomial P(x) = ax 2 + bx + 2a – 3b – 9 so
that P(1) = 0 and P(2) = 0.
(2005 – 2nd Session)
There are some missing numbers in the following text:
« For buying….. Pencils and 2 pens we pay ….. L L, and for buying ….. Pencils and 3 pens we pay
7800 L L. »
24
Setting up the complete given of the text in equations give the following system: {
4x + 2y = 5600
2x + 3y = 7800
1) Copy again the complete text according to the given system.
2) Solve, showing all the steps of calculation, the preceding system and find the price of one pencil and
the price of one pen.
(2006 – 1st Session)
7
x
, we get a fraction equal to . If we subtract 3 from
8
y
3
each term of this fraction, we get a fraction equal to .
4
1) Show that the preceding information is translated into the following system of two equations with two
8𝑥 − 7𝑦 = −5
unknowns: {
4𝑥 − 3𝑦 = 3
x
2) Solve this system, showing all the steps of calculation, and find the fraction .
y
If we add 5 to each term of the irreducible fraction
(2006 – 2nd Session)
To buy two copybooks and one pen we must pay 2750 LL, and to buy four copybooks and three pens we
must pay 7750 LL. The preceding information is translated into the following system:
 2x  y  2750

4x  3y  7750
1) What does x and y represent in this system?
2) Which information is translated by the equation 4x  3y  7750 ?
3) Solve the preceding system, showing the followed steps in detail, to find the price of a copybook and
the price of a pen.
(2007 – 1st Session)
In what follows, designate by 𝑥 the price of a pen in L L and 𝑦 the price of a copybook in L L.
To buy one pen and one copybook we pay 2500 L L. If the price of a pen is decreased by 30% and the
price of a copybook is decreased by 20% the amount we pay becomes 1900 L L.
x  y  2500
1) Prove that the preceding information is translated into the following system : 
7 x 8y  19000
2) Solve the preceding system, showing the followed steps in detail, and find the price of one pen and the
price of one copybook.
(2008 – 2nd Session)
A person bought a computer for 1 200 000 LL. He paid one quarter of this price as a down-payment.
The remaining amount of the price is to be paid in 10 monthly payments:
Some of these payments are 150 000 LL each, and the others are 50 000 LL each.
Let x be the number of the 150 000 LL payments.
1) Calculate the down-payment, and the remaining amount of the price of this computer.
2) Calculate the number of the 150 000 LL payments, and that of the 50 000 LL payments.
25
(2009 – 1st Session)
A first bunch of flowers is formed by 3 roses and 4 tulips and it costs 4800 LL.
A second bunch of flowers is formed by 5 roses and 6 tulips and it costs 7500 LL.
Denote by x the price of one rose, and by y the price of one tulip.
1) Write a system of two equations modeling the previous information.
2) Solve the previous system in showing the steps of calculation. Determine the price of one rose and
that of one tulip.
3) A client buys a bunch formed by 10 flowers and he pays 6450 LL.
Calculate the number of roses and that of tulips in this bunch.
(2010 – 2nd Session)
A bookshop offers a 10% discount on its articles.
The sum of original prices of a pen and an agenda is three times the original price of the pen.
The sum of prices of the pen and the agenda after discount is 54000 LL.
1) Model the previous information into a system of two equations with two unknowns.
2) Solve this system and find the original price of a pen and that of an agenda.
(2011 – 2nd Session)
To buy three copybooks and two pens we must pay 4 500 LL. To buy six copybooks and three pens we
must pay 7 500 LL. This information is translated into the following system:
3x + 2y = 4 500
{
6x + 3y = 7 500
1) What does x and y represent in this system?
2) Solve the previous system, showing the details of the steps you follow, to find the price of a copybook
and that of a pen.
3) A student bought a pack which contains copybooks and pens, and he paid 11 000 LL. Calculate the
number of copybooks and the number of pens in this pack knowing that the sum of these two numbers
is 12.
(2014 – 1st Session)
x + y = 120 000
4x + 5y = 500 000
2) A store offers 40% discount on the price of pants and 25% on the price of shirts.
The sum of the original prices of a pant and a shirt is 120 000 LL, while their sum is 75 000 LL after
the discount.
Denote by x the original price of a pant and by y that of a shirt.
a. Express, in terms of x and y, the new prices after the discount.
b. Model the previous information into a system of two equations with two unknowns.
c. What is the original price of a pant and that of a shirt?
1) Solve the following system: {
(2015 – 1st Session)
x + y = 35
2x − 3y = 0
2) Find, with justification, two natural numbers such that their sum is 35 and the double of one of them is
triple of the other.
1) Solve the following system: {
26
(2016 – 1st Session)
6x + 4y = 20 000
2x + 8y = 15 000
2) A bookshop offers 40% discount on the price of a copybook and 60% discount on that of a pencil.
The sum of the original prices of 2 copybooks and 8 pencils is 15 000 L.L.
The sum of prices, after discount, of one copybook and one pencil is 2 000 L.L.
a. Prove that the previous information can be modeled by the above system.
b. Find the price of a copybook and that of a pencil after the discount.
1) Solve the following system: {
(2017 – 1st Session)
x + y = 35
9x + 8y = 300
2) The number of students (girls and boys) of a certain class is 35.
When 10% of the girls and 20% of the boys leave this class to participate in a sportive activity, the
number of remaining students is then 30.
a. Denote by x the number of girls and by y that of boys of this class.
Write a system of two equations with two unknowns to model the text above.
b. Find the number of girls and that of boys in this class.
1) Solve the following system: {
(2017 – 2nd Session)
5x + 2y = 12 000
x + 2y = 8 000
2) A restaurant sells 10 green salads and 4 vegetarian pizzas for 24 000 L.L.
The same restaurant sells 6 green salads and 12 vegetarian pizzas for 48 000 L.L.
Show that this text is modeled by the system given in question 1)
3) Nadine orders 8 green salads and 6 vegetarian pizzas, how much will she pay?
1) Solve the following system: {
(2018 –1st Session)
1) Solve the following system: {
2x + 5y = 50 000
2x + 3y = 38 000
2) In a museum, 2 adults and 5 kids buy tickets and pay 50 000 LL;
4 adults and 6 kids pay 76 000 LL.
a. Prove that the previous information is modeled by the system given in question 1).
b. Find the price of the ticket of an adult and that of a kid.
3) For a group of 30 kids and 4 adults, the director of the museum decided to offer a reduction of 25% on
the total amount paid for the tickets. Calculate then the amount paid.
(2018 – 2nd Session)
1) Solve, showing all the steps of calculation, the following system: {
x − 2y = 0
3y − x = 6
2) In a class, the number of boys is double that of girls.
If 2 girls leave the class, the number of boys becomes triple that of the girls.
The teacher confirms that there are 18 students in this class. Is he right? Justify.
27
(2019 –1st Session)
 x  y  16
1) Solve the following system: 
.
2x  3y  38
2) The following table represents the distribution of electronic games in a shop according to their prices:
Price of an electronic game (in LL)
Number of electronic games
3 000
9
4 000
m
5 000
15
6 000
n
a. The total price of all electronic games in this shop is 178 000 LL.
Show that this information is modeled by the following equation: 2m + 3n = 38.
b. Knowing that the total number of electronic games in this shop is 40. Calculate m and n.
3) Given: m = 10 and n = 6.
Calculate the average price (mean) of these 40 electronic games.
(2019 – 2nd Session)
A box F contains twelve red and black balls.
1) If one red ball is removed and one black ball is added, then the number of red balls becomes double
that of black balls.
 x  y  12
a. Prove that the previous information is modeled by the following system: 
.
 x  2y  3
b. Solve the previous system and determine the number of red balls and that of black balls.
2) In what follows, the box F contains nine red balls and three black balls.
Five red balls and eight black balls are added to this box.
Calculate the percentage of red balls in this box.
(2019 – 3rd Session)
 x  y  12
1) By showing all the steps of calculation, solve the following system: 
.
3x  4y  44
2) A bag contains 12 chocolate bars of 2 types M and T. The price of this bag is 22 000 LL.
Each bar of type M costs 1 500 LL and that of type T costs 2 000 LL.
a. Prove that the previous information is modeled by the system in question 1).
b. Find the number of chocolate bars of each type.
3) If the price of a bar of chocolate of type M is increased by 20% and that of chocolate of type T is
decreased by 10%, does the price of this bag change? Justify.
28
Lines in coordinate system – Analytic geometry
(2001 – 1st Session)
In an orthonormal system of axes x′Ox, y′Oy, given the points A (-1; 0); B (0; 2) and E (–3; -2).
1) Locate A, B, E in this system.
2) a. Prove that: 𝐴𝐸 = √20 , 𝐴𝐵 = √5 , 𝐵𝐸 = 5.
̂B
b. Prove that triangle ABE is right angled at A and calculate cos AE
3) Designate by (C) the circumscribed circle of triangle ABE and by I the center of this circle.
a. Find the coordinates of I and calculate the radius of (C).
b. Determine the coordinates of points of intersection, other than A and B, of circle (C) with the axes
x′Ox and y′Oy.
(2001 – 2nd Session)
9
1
In an orthonormal system x′Ox, y′Oy, given the points: R (1; 2) , S(2; 2), T(−4; −3), U (−5; − 2) and
3
the straight line (D) of equation 𝑦 = 2 x + 3
1) Verify, by calculation, that R and T are two points of the straight line (D).
2) Plot the points R, S, T, U and draw the straight line (D).
3) a. Calculate the coordinates of point I, midpoint of the segment [RT].
b. Verify that I is the midpoint of the segment [SU].
c. What is the nature of the quadrilateral URST? Justify the answer.
(2002 – 1st Session)
In an orthonormal system of axes x′Ox, y′Oy, given the points A (6; 5); B (2; –3), C (– 4; 0) and the
straight line (d) of equation y = 2x – 7.
1) Plot the points A, B and C.
2) a. Prove by calculation, that (d) passes through the points A and B. Then draw (d).
b. By using the director coefficient of (d), calculate the acute angle that (d) makes with x′Ox.
3) Prove that AB = 4√5 , BC = 3√5 and CA = 5√5 .
4) a. Prove that triangle ABC is right angled at B.
b. Let I be the center of the circle circumscribed about triangle ABC.
Calculate the radius of this circle and find the coordinates of its center I.
5) Let F be the image of point C by the translation of vector ⃗⃗⃗⃗⃗
BA.
a. Calculate the coordinates of vector ⃗⃗⃗⃗⃗
BA .
b. Calculate the coordinates of point F.
c. What is the nature of quadrilateral CBAF? Justify your answer.
(2002 – 2nd Session)
In the plane of an orthonormal system x′Ox, y′Oy, consider the three points A(2; 5); B(6; 5) ; C(1 ; 0)
and the line (D) of equation x = 4.
1) Plot A, B and C.
2) a. Draw the line (D).
b. Prove that (D) is the perpendicular bisector of the segment [AB].
29
3) Show that the equation of the line (BC) is y = x – 1.
4) Let E be the midpoint of the segment [BC].
a. Calculate the coordinates of E.
b. Show that the equation of line (D′), the perpendicular bisector of [BC], is y = – x + 6.
5) Let H be the point of intersection of (D) and (D′), and let (S) be the circle with center H and passing
through A.
a. Prove that the circle (S) passes through the points B and C.
b. Calculate the coordinates of H.
c. Calculate the radius of circle (S).
(2003 – 1st Session)
The unit of length is the cm and the unit of area is the cm2.
In the opposite figure, EFGB is a rectangle and CEF is a triangle, right angled at F.
Given that GB = 6, GF = 3 and FC = 5.
The point M moves on the segment [EF].
Let x be the length of MF such that0 ≤ 𝑥 ≤ 6.
E
Part A:
1) Express, in terms of x, the area A1 of the triangle FCM.
C
M
x
F
3
2) Let A2 be the area of the triangle MEB. Prove that: A2 = − x + 9
2
3) For what values of x, is the area A1 strictly greater than the area A2?
G
B
Part B:
5
Given in the plane of an orthonormal system of axes x′Ox, y′Oy, the line (d1) of equation y = x and the
2
3
line (d2) of equation y = − 2 x + 9
1) Draw (d1) and (d2).
2) a. Calculate the coordinates of the point K the intersection point of the two lines (d1) and (d2).
b. Deduce the numerical value of x so that the two triangles FCM and MEB have the same area.
(2003 – 1st Session)
In the plane of orthonormal system x′Ox, y′Oy, Given the points A(-2; 1); B(3; 6), C(4 ; -1) and D(-1,-6).
1) Plot the points A, B, C and D.
2) Calculate the coordinates of point E the midpoint of [AC] and verify that E is the midpoint of [BD].
3) Prove that the quadrilateral ABCD is a rhombus.
4) Calculate the radius of the circle circumscribed about the triangle AEB.
(2003 – 2nd Session)
In the plane of an orthonormal system of axes x′Ox, y′Oy, consider the straight line (d) of equation y =
2x + 4 and the points E(– 1; – 4); H(– 1; 2), K(2; – 4) and S(– 4 ; 2).
1) Draw (d) and verify by calculation that H is a point on (d).
2) The line (d) cuts the x-axis in a point M and the y-axis in N. Prove that ⃗⃗⃗⃗⃗⃗
MH = ⃗⃗⃗⃗⃗⃗
HN .
30
3) Write the equation of the straight line (EH) and the equation of the straight line (EK).
4) a. Prove that triangle HEK is a right triangle.
b. Calculate the radius of the circle circumscribed about triangle HEK.
⃗⃗⃗⃗⃗ .
5) Prove that S is the image (translate) of E by the translation of vector KH
6) Designate by (d′) the image (translate) of (d) by the translation of vector ⃗⃗⃗⃗⃗
NO .
Draw (d′) and write its equation.
(2004 – 1st Session)
Consider in an orthonormal system of axes x Ox , yOy , the points A(4 ; 2) , B(-2 ; -2) and the line (d)
of equation y = – x + 4 .
1) Draw (d) and plot A and B.
2) Calculate the coordinates of point G the midpoint of segment [OA].
3) a. Determine the equation of the straight line (OA).
b. Let (  ) be the perpendicular bisector of segment [OA].Show that the equation of (  ) is y= – 2x+ 5
4) Let M be the intersection point of the two straight lines (  ) and (d).
a. Justify that MO = MA.
b. Calculate the coordinates of M.
c. Prove that the triangle MOA is a right isosceles triangle.
⃗⃗⃗⃗⃗ . Prove that NB = MA.
5) Designate by N the image (translate) of M by the translation of vector OB
(2004 – 2nd Session)
In an orthonormal system of axes x’Ox, y’Oy, consider the point C (0; 3) and the straight line (D) of
1
equation y = x – 2.
2
1) (D) cuts x’Ox in A and y’Oy in B. Calculate the coordinates of A and B, and draw (D).
2) The perpendicular (D') drawn from C to (D) cuts the straight line (D) in I.
a. Find the equation of (D').
b. Calculate the coordinates of I.
⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
3) Let E be the point such that AE
AB + AC
a. What is the nature of quadrilateral ABEC?
b. Calculate the coordinates of E.
4) Let M (0; m) be a point on y’Oy, where m is a positive number.
a. Calculate the numerical value of m such that triangle ABM is right at A.
b. For this value of m, the circle of diameter [MB] cuts again the axis x’Ox in a point H.
What are the coordinates of H? Justify.
31
(2005 – 1st Session)
Consider in an orthonormal system of axes x O x , y O y, the points :A(– 3 ; 3), B(2 ; – 2), G(– 4 ; – 2)
and E(2 ; 2).
1) Plot the points A, B, G and E.
2) a. Justify that the straight line (BE) is parallel to (y y) and that (BG) is parallel to (x x).
b. Prove that the triangle BGE is right angled at B.
̂ E.
c. Calculate tan 𝐵𝐺̂ 𝐸 and calculate, rounded to the nearest degree, the angle BG
3) Designate by (C) the circle circumscribed about triangle BGE.
Prove that its center is the point I( – 1 ; 0), and calculate the exact value of its radius.
4) Prove that A is a point of the circle (C).
5) a. Find the equation of the straight line (GE).
b. Prove that (GE) and (AI) are perpendicular.
⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗ . Prove that the quadrilateral AGFE is a square.
c. Let F be the point such that AE
AG = AF
(2005 – 2nd Session)
In an orthonormal system of axes x O x , y O y, consider the points :A(– 4 ; 4), B(3 ; 3) and C(1 ; – 1)
1) Plot the points A, B and C.
2) Prove that the three points A, O and C are collinear.
3) Prove that the triangle ABC is isosceles of principal vertex A.
4) Let H be the midpoint of [BC]. Prove that (AH) is perpendicular to (BC).
5) Let N be the image (translate) of B by the translation of vector ⃗⃗⃗⃗⃗
AC. Prove that CABN is rhombus.
(2005 – 2nd Session)
In the opposite figure where the unit of length is the
centimeter :
 ABC is a triangle right angled at A
 AB = 8 and AC = 4
 M is a point of [AB] such that:
BM = x and 0  x  8
 (ME) is perpendicular to (AB).
C
E
Part A:
1
x.
2
2) Calculate x so that the triangle AME is isosceles.
1) Prove that: ME =
A
B
M
Part B:
Consider an orthonormal system of axes x  O x , y  O y.
1) Draw, in this system, the two straight line (d) of equation y =
equation y =  x + 8.
2) Using the graph, find again the result of question 2) of part A.
32
x
2
and the straight line (d ) of
Part C:
1) Calculate the exact value of the length of side [BC] of triangle ABC.
2) Write the value of BC appearing on your calculator, then give this value rounded to the nearest 10 2
by default.
̂ C, then calculate, rounded to the nearest degree, the angle AB
̂ C.
3) Calculate tan AB
(2006 – 1st Session)
Consider in an orthonormal system of axes x Ox , yOy , the points A(– 3 , 6), B(5 , 2) and E(1,– 2) and
the straight line (d) of equation y = x – 3 .
1) Plot the points A, B and E.
2) Verify by calculation, that E and B are two points of straight line (d). Draw (d).
3) a. Write the equation of the straight line (AE). Deduce that the points E, A and O are collinear.
b. Are the straight lines (d) and (AE) perpendicular? Justify your answer.
4) Designate by M and N the respective images (translates) of A and C by the translation of vector ⃗⃗⃗⃗⃗
EB,
and designate by (D) the image (translate) of y' y  by the same translation.
a. Prove that B, N and M are collinear.
⃗⃗⃗⃗⃗ .
b. Calculate the coordinates of EB
c. Calculate the coordinates of N.
d. Draw (D) and find its equation.
5) a. Show that AEBM is a parallelogram and not a rectangle.
b. The diagonals of AEBM intersect in J. Calculate the coordinates of J.
(2006 – 2nd Session)
Consider in an orthonormal system of axes x Ox, y Oy, the points: A 2; 2 ; B 3; 1 and E 0;  1
'
'
1) Plot the points A, B and E.
2) Write an equation of the line (BE).
3) Knowing that AB  26 and BE  13 , calculate AE and prove that the triangle ABE is an
isosceles right triangle at E.
4) Let (C) be the circle circumscribed about triangle ABE. Calculate the radius of (C) and the
coordinates of its center J.
5) Designate by F the image (translate) of A by the translation of vector ⃗⃗⃗⃗⃗
EB.
a. Prove that AEBF is a square.
b. Deduce that F is a point of (C).
c. Calculate the coordinates of F.
(2007 – 1st Session)
In the plane of an orthonormal system x O x, y O y, where the unit of length is the centimeter, consider
3
the straight line (d) of equation y =  x – 1 and the points A(– 4 ; 5), B(6 ; 3) and G(0, –1).
2
1) Plot the points A, B and G.
2) Verify by calculation, that A and G are two points of (d), then draw (d).
3) Write an equation of the straight line (BG) and deduce that the straight lines (d) and (BG) are
perpendicular.
33
4) Knowing that AG = 2 13 . Calculate BG and deduce that AGB is an isosceles right triangle.
5) Let (C) be the circle circumscribed about triangle ABG. Calculate the radius of (C) and the
coordinates of its center J.
⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
6) Designate by E the point defined by GE
GA + GB
a. Prove that AGBE is a square.
b. Calculate the coordinates of E.
c. Prove that E is a point of (C).
(2007 – 2nd Session)
Consider, in an orthonormal system of axes x Ox and yOy where the unit of length is the centimeter,
1
the points A(0 ; – 4) , E(0 ; 1) , F(4 ; – 1) and the straight line (d) of equation y   x  1.
2
1) Plot the points A, E and F.
2) Verify by calculation, that E and F are two points of (d), and then draw (d).
3) Prove that I(2 ; 0) is the midpoint of [EF].
4) We know that EF  2 5 .
a. Calculate AE and AF. Deduce that triangle AEF is isosceles of principal vertex A.
b. Is the straight line (AI) perpendicular to (EF)? Justify.
5) Let B be the symmetric of A with respect to I.
a. Prove that AFBE is a rhombus.
b. Calculate the coordinates of B.
6) Let (d') be the straight line passing through B and parallel to (d). Determine the equation of (d').
7) (AE) and (AF) intersect (d') in M and N respectively. Prove that EMNF is an isosceles trapezoid and
calculate its area.
(2008 – 1st Session)
In an orthonormal system of axes (x'ox, y'oy) , Consider the points E(3;3) , F(2; 2) and G(2;4).
1) Locate the points E, F and G.
2) a. Given that EG = 26 . Calculate EF and FG.
b. Deduce that the triangle EFG is isosceles and right angled at E.
3) Let (C) be the circle circumscribed about triangle EFG.
a. Find the radius R of (C).
b. Calculate the coordinates of the point I, the center of (C), and deduce that I is on  y ' y  .
c. Show that P(3,3) belongs to the circle (C).
⃗⃗⃗⃗⃗ .
4) a. Calculate the coordinates of point L, the translate (image) of E under the translation of vector OP
b. Determine the equation of the straight line (OE).
c. Determine the equation of the straight line (d'), the translate (image) of (OE) under the translation
of vector ⃗⃗⃗⃗⃗
OP.
d. Show that P, G and L are collinear.
34
(2008 – 2nd Session)
In
the
orthonormal
system
(D1) of equation y = 3x + 6
and
of
axes
x'Ox,
y’Oy,
x
(D2) of equation y = –
+ 3.
3
given
the
two
lines:
1) Plot the two lines (D1) and (D2).
2) Prove that (D1) and (D2) are perpendicular.
3) The two lines (D1) and (D2) intersect at A. Calculate the coordinates of the point A.
4) Plot the point C(9; 0) and show that C is on (D2).
5) (D1) cuts x'Ox at a point B. Calculate the coordinates of B.
6) Let E be the point of intersection of (D1) and y'Oy.
a. Find the coordinates of the point E.
b. Calculate the coordinates of the center M of the circle circumscribed about the triangle AEC.
c. Find an equation of the straight line (∆) the translate of (D2) under the translation with vector ⃗⃗⃗⃗⃗
AE.
(2009 – 1st Session)
Consider in an orthonormal system of axes x′Ox, y′Oy the line (d) with equation y = 3x + 2 and the two
points A (1; 5) and B(– 2 ; – 4).
1) Show that the points A and B belong to the line (d).
2) Locate A and B and plot (d).
3) Let (d′) be the perpendicular bisector of [AB] and H the midpoint of [AB].
Calculate the coordinates of H, and then determine the equation of (d′).
4) Let M(– 5; 2) be a point on (d′).
a. Given that MA =3√5 , justify that MB = 3√5 .
b. Calculate AB and deduce that AMB is a right isosceles triangle.
5) Consider the point P on the line (d′) so that P is distinct from M and AP = AM.
a. Locate P, and show that BP = BM.
b. What is the nature of the quadrilateral MAPB? Justify.
(2009 – 2nd Session)
In an orthonormal system of axes x′Ox, y′Oy with 1 cm as unit of length, consider the points A (– 2; 3),
B(1; – 1), C(9; 5) and the line (d) with equation y = 2x – 13.
1) Locate the points A, B, C and plot the line (d).
2) Calculate the coordinates of N the intersection point of (d) with the axis x′Ox.
3) Prove that the triangle ABC is right angled at B.
4) Let M be the midpoint of [AC], calculate the coordinates of M.
5) Show that N is the translate of C by the translation with vector ⃗⃗⃗⃗⃗⃗
MB.
6) Prove that the quadrilateral BMCN is a rhombus.
(2010 – 1st Session)
In an orthonormal system of axes x′Ox and y′Oy, consider the points A (1; 1) and B (3; 2).
1) Locate the points A and B and calculate AB.
1
1
2) Verify that the equation of (AB) is y = 2 x + 2
35
3) The circle (C) with center A and radius AB intersects (AB) at another point D.
Calculate the coordinates of D and deduce that D is on the axis x′Ox.
4) Let (d) be the tangent at B to the circle (C).
a. Determine the equation of (d).
b. Verify that (d) intersects y′Oy at the point E (0; 8).
c. Calculate the coordinates of F, the intersection point of (d) and the axis x′Ox.
5) Let H be the translate of E by the translation with vector ⃗⃗⃗⃗⃗
BA . Calculate the coordinates of the point H.
6) What is the nature of the quadrilateral ABEH? Justify.
̂ B rounded to the nearest degree.
7) Calculate the value of the angle EA
(2010 – 2nd Session)
In the plane referred to an orthonormal system of axes x′Ox and y′Oy, consider the line (d) with equation
y = – 2x + 3 and the points A (0; – 2) and E (6; 1).
1) Locate A and E.
2) The line (d) intersects y′Oy at G.
a. Calculate the coordinates of G.
b. Locate G and plot (d).
1
3) a. Show that y = 2 x − 2 is the equation of (AE).
b. Prove that (d) is perpendicular to (AE) at the point B (2; – 1).
c. Prove that the right triangle GBE is isosceles.
4) Denote by I the midpoint of [GE] and by M the symmetric of B with respect to I.
a. Prove that the quadrilateral BGME is a square.
b. Calculate the coordinates of M.
⃗⃗⃗⃗⃗ .
5) (d′) is the translate of the line (AE) by the translation of the vector BG
a. Plot (d′).
b. Find the equation of (d′).
(2011 – 1st Session)
In an orthonormal system of axes x′Ox and y′Oy, consider the line (D) with equation y = – 2x – 3 and the
two points A (– 2; 1) and B(6 ; 5).
1) Verify that (D) passes through A.
2) Plot A and B and draw (D).
3) Determine the equation of (AB) and deduce that (D) is perpendicular to (AB).
4) Calculate, rounded to the nearest degree, the value of the acute angle that (AB) makes with x'Ox.
5) The line (D) intersects y'Oy at C. Find the coordinates of C.
6) Let (S) be the circle circumscribed about the triangle ABC & I is the center of this circle.
Calculate the coordinates of I.
7) The line (D') is the parallel through C to (AB). (D') intersects the circle (S) at another point E.
a. What is the nature of the quadrilateral ABEC? Justify.
b. Calculate the coordinates of the point E.
c. (d) is the tangent at A to (S). Find the equation of (d).
36
(2011 – 2nd Session)
In an orthonormal system of axes x′Ox and y′Oy, consider
A (3; 4) , B(3 ; – 1) & C(1; 3) and the line (d) with equation y = – 2x + 5.
1) Plot the points A, B and C.
2) Prove that B and C are two points on (d), then draw (d).
3) a. Find the equation of the line (CA).
b. Prove that the triangle ABC is right at C.
4) D is the point defined by ⃗⃗⃗⃗⃗
CD = ⃗⃗⃗⃗⃗
CA + ⃗⃗⃗⃗⃗
CB.
the
points
points:
a. Prove that CADB is a rectangle.
b. Calculate the coordinates of D.
5) E is the symmetric of C with respect to A.
a. What is the nature of the quadrilateral ABDE? Justify.
b. Prove that CDE is an isosceles triangle.
c. Prove that (DE) is parallel to y'Oy and write the equation of (DE).
(2012 – 1st Session)
In an orthonormal system of axes x′Ox and y′Oy, where the unit of length is the centimeter, consider the
points A (1; –2), B(2 ; 1) C(5; 0) and line (d) with equation y = 3x – 5.
1) a. Verify that (d) is passing through A and B.
b. Plot the points A, B, C and draw (d).
2) a. Determine the equation of the line (BC).
b. Calculate the lengths AB and BC.
c. Prove that ABC is right isosceles triangle with vertex B.
3) Let (G) be the circle circumscribed about the triangle ABC and D is the point defined by ⃗⃗⃗⃗⃗
BD = ⃗⃗⃗⃗⃗
AC
a. Calculate the coordinates of D.
b. Prove that (DB) is tangent to (G).
⃗⃗⃗⃗⃗ . Find the equation of (d').
4) (d') is the translate of (d) under the translation with vector AC
(2012 – 2nd Session)
In an orthonormal system of axes x′Ox and y′Oy, consider the points A (–1; 0); B(0; 2) and E (3; –2).
1) Plot A, B, and E in this system.
2) a. Prove that BE = 5.
b. Let I be the midpoint of [BE]. Calculate the coordinates of I.
c. Calculate AI and deduce that the triangle ABE is right at A.
3) Denote by (C) the circle circumscribed about triangle ABE and by (t) the tangent at B to (C).
4
a. Verify that the slope of (BE) is equal to − 3
b. Write the equation of (t).
c. (t) intersects x'Ox at F. Calculate, rounded to the nearest degree, the measure of the angle BF̂I.
37
(2013 – 1st Session)
In an orthonormal system of axes x′Ox and y′Oy, where the unit of length is the centimeter, consider the
line (d) with equation y = – x – 4 and the points A (–1; –3); B (–7; 3) and C (3; 1).
1) a. Verify that A and B are two points on (d).
b. Plot the points A, B and C. Draw (d).
2) a. Calculate BC.
b. Knowing that AB = 6√2 and AC = 4√2 , prove that ABC is a right triangle.
3) Let J be the center of the circle circumscribed about the triangle ABC. Calculate the coordinates of J.
4) Prove that the line (d') with equation y = x + 4 is the perpendicular bisector of [AB].
5) a. Calculate the coordinates of the vector ⃗⃗⃗⃗⃗
BC.
⃗⃗⃗⃗⃗ .Calculate the coordinates of D.
b. Let D be the translate of A under the translation with vector BC
6) Let A' be the symmetric of A with respect to J.
a. Prove that ABA'C is a rectangle.
b. Prove that C is the midpoint of [DA'].
(2013 – 2nd Session)
In an orthonormal system of axes x′Ox and y′Oy, consider the points A (–2; –2); E (2; 6) and B (6; –2).
1) a. Plot A, B, and E.
b. Verify that the equation of (AE) is y = 2x + 2.
2) Determine the equation of (AB).
3) Verify that AE = BE.
4) Let K be the midpoint of the segment [AE].
a. Calculate the coordinates of K.
b. Let (d) be the perpendicular to (AE) at K. Determine the equation of (d).
5) The line (d) intersects the perpendicular bisector of [AB] at I.
a. Show that I is the center of circle circumscribed about the triangle ABE.
b. Calculate the coordinates of I.
6) Let J be the symmetric of E with respect to I. Show that (AJ) is parallel to (d).
(2014 – 1st Session)
In an orthonormal system of axes x’Ox; y’Oy, consider the line (d) with equation y = -2x + 6 and the
points E (6; 4) and F (0; 1).
1) Plot the points E and F. Draw (d) and determine the coordinates of B, the intersection point of (d) and y’Oy.
1
2) Let (d’) be the line passing through E and F. Show that 𝑦 = 2 𝑥 + 1 is the equation of (d’).
3) a. Verify that P (2; 2) is the intersection point of (d) and (d’).
b. Calculate PB and PE.
c. Show that BPE is a right isosceles triangle.
4) Let M be the midpoint of [BE].
a. Calculate the coordinates of M.
̂E.
b. Determine the equation of the bisector of the angle BP
38
(2015 – 1st Session)
In an orthonormal system of axes x’Ox; y’Oy, consider the points A (3; 3), B (0; – 3) and C (– 6; 0).
1) Plot the points A, B and C.
2) Verify that y = 2x – 3 is the equation of the line (AB).
3) Calculate the slope of the line (BC). Deduce that (AB) and (BC) are perpendicular.
4) Show that ABC is a right isosceles triangle.
⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗
5) Let D be the point defined by AD
BC.
a. Verify that the coordinates of D are (–3; 6).
b. Show that the quadrilateral ABCD is a square.
6) Let E be the symmetric of D with respect to A and (G) the circle circumscribed about triangle CDE.
a. Calculate the coordinates of E.
b. Calculate the coordinates of I the center of circle (G).
c. Determine the equation of the tangent at D to the circle (G).
(2015 – 2nd Session)
In an orthonormal system of axes x’Ox; y’Oy, consider the points A (0; 2) and B (– 4; 0).
1) Plot the points A and B.
1
2) Show that y = 2 x + 2 is an equation of the line (AB).
3) Let [OH] be an altitude in triangle OAB.
a. Find the equation of line (OH).
4 8
b. Verify that the coordinates of H are (− 5 ; 5)
4) The parallel through B to y’Oy intersects (OH) at E.
a. Calculate the coordinates of point E.
b. Calculate OE and HE.
5) Let (C) be the circle circumscribed about triangle OBE and (d) the tangent at O to (C).The two lines
(d) and (EA) intersect at F. Prove that
EA
EF
=
4
5
(2016 – 1st Session)
In an orthonormal system of axes(x′Ox, y′Oy), consider the points A (–2; 0) and B (1; 3). Let (d) be the
line with equation y = – x + 4.
1) a. Plot the points A and B.
b. Verify, by calculation, that point B is on the line (d), then draw (d).
c. Determine the equation of line (AB) and verify that (AB) is perpendicular to (d).
d. The line (d) intersects x'Ox at E and y'Oy at F. Calculate the coordinates of points E and F.
2) Let (C) be the circle circumscribed about the triangle ABF.
a. Determine the coordinates of point I, the center of (C) and calculate the radius of this circle.
b. Verify that O is a point on the circle (C).
3) a. Calculate AB.
̂ F.
b. Calculate, rounded to the nearest degree, the measure of BA
39
(2016 – 2nd Session)
In an orthonormal system of axes(x’Ox; y’Oy), consider the line (d) with equation y = −2x + 3 and the
points A (0; – 2), E (6; 1) and G (0; 3).
1) Plot the points A, E and G.
2) Verify that G is a point on line (d), then draw (d).
1
3) a. Show that y = 2 x − 2 is the equation of line (AE).
b. Prove that the lines (d) and (AE) are perpendicular.
c. Verify by calculation, that B (2; – 1) is the intersection point of lines (d) and (AE).
d. Prove that GBE is a right isosceles triangle.
4) Denote by M the translate of E under the translation with vector ⃗⃗⃗⃗⃗
BG.
a. Prove that the quadrilateral BEMG is a square.
b. Calculate BM.
(2017 – 1st Session)
In an orthonormal system of axes x’Ox and y’Oy, consider the points A(-2; 2) ; B(0; -2) ; C(5; 3) and
1
1
I(-1; 0). Let (d) be the line with equation y = 2 x + 2.
1) a. Plot the points A, B, C and I.
b. Verify that C and I are two points on the line (d). Draw (d).
2) Prove that I is the midpoint of [AB].
3) a. Find the equation of the line (AB)
b. Prove that (AB) is perpendicular to the line (d).
c. Show that the triangle ABC is isosceles.
4) Consider the point F (7; -1). Show that F is the translate of C under the translation with vector ⃗⃗⃗⃗⃗
𝐴𝐵 .
5) Denote by E the point on the line (AB) so that XE = 1.
a. Show that YE = −4.
b. Prove that the quadrilateral CIEF is a rectangle.
(2017 – 2nd Session)
In an orthonormal system of axes(x’Ox; y’Oy), Consider the points A(4; 2), B(−1; 2) and E(1; 3).
Let (d) be the line with equation y = 3x.
1) a. Plot A, B and E.
b. Verify that E is a point on the line (d). Draw (d).
2) a. Calculate OB and show that OA=2OB.
b. Show that OAB is a right angled triangle.
3) a. Determine the coordinates of point I, the symmetric of O with respect to B.
b. Verify that E is the midpoint of [AL].
4) Let (d′ ) be the line passing through A and perpendicular to (𝑂𝐴).
Show that the equation of (d′ ) is y = −2x + 10.
5) Let F be the point with coordinates (2; 6).
a. Verify that F is the intersection point of (d) and (d′ ).
b. Prove that the quadrilateral OAFL is a square.
40
(2018 –1st Session)
In an orthonormal system of axes x'Ox and y'Oy, consider the points A1; 0and B1; 4
Let (d) be the line with equation y = 2x + 2.
1) a. Verify that A and B are two points on line (d).
b. Plot the points A and B then draw (d).
2) Let I be the point of intersection of (d) with y'Oy.
a. Calculate the coordinates of I.
b. Verify that I is the midpoint of [AB].
1
3) Let (d') be the perpendicular bisector of [AB].Verify that the equation of (d') is y = 2 x + 2.
4) Consider the point M(4 ; 0). Show that triangle MAB is isosceles of vertex M.
5) Let K be the translate of B under the translation with vector ⃗⃗⃗⃗⃗⃗
MA.
Prove that quadrilateral MBKA is a rhombus.
(2018 – 2nd Session)
In an orthonormal system of axes x'Ox and y'Oy, given the points F; 4and B2; 2
Let (d) be the line with equation y = x + 4.
1) Plot the points F and B.
2) Show that F and B are two points on (d), then draw (d).
3) Let H be the point of coordinates 1; 3).
a. Verify that H is the midpoint of [BF].
b. Show that the equation of (d′ ), the perpendicular bisector of [BF], is y = −x + 2.
4) a. Show that (OB) and (d′ ) are parallel.
b. Show that the triangle OBF is right isosceles at B.
5) Let (C) be the circle circumscribed about triangle OBF.
Show that the point E; 2is the center of (C), and calculate its radius.
6) Let K be the point of coordinates and F; 4the point of intersection of (d′ ) and x'Ox.
Show that K is a point of circle (C), and that (LK) is tangent to circle (C).
(2019 –1st Session)
In an orthonormal system of axes x 'Ox et y' Oy, given the points A (0 ; − 2), B (− 4 ; 0) and C (0 ; 3).
1) a. Plot the points A, B and C.
1
x2.
2
2) Show that the triangle ABC is isosceles of vertex C.
3) Let H be point with coordinates (− 2; − 1).
a. Verify that H is the midpoint of [AB].
b. Determine the equation of the bisector of angle BĈA.
b. Show that the equation of line (AB) is y 
41
3

4) a. Prove that the points B, H, O and C are on the same circle of center I  2 ;  and calculate its
2

radius.
b. Show that (IH) is parallel to the y-axis.
5) Let K be the point of coordinates (− 2; 0). Calculate the area of trapezoid HACI.
(2019 – 2nd Session)
In an orthonormal system of axes x′Ox and y′Oy, given the points A (3; 3), B (6; 0) and E (0; − 6).
Let (d) be the line with equation y = − x + 6.
1) a- Plot the points A, B and E.
b- Verify that A and B are two points on (d). Draw (d).
2) The line (d) intersects y′Oy at F.
Determine the coordinates of F, then verify that A is the midpoint of [BF].
3) a- Verify that an equation of the line (AE) is y = 3x – 6.
b- The line (AE) intersects x′Ox at C (2; 0). What does point C represent for triangle EBF?
c- The lines (CF) and (BE) intersect at M. Prove that M is the midpoint of [BE].
̂ F = OB
̂ E = 450 .
4) Prove that OB
5) The parallel through C to (EB) intersects [FB] at K.
a- Show that the triangle CKB is right isosceles of vertex K.
b- Deduce that CK = 2 2 .
FC
6) Calculate the ratio
.
FM
(2019 – 3rd Session)
In an orthonormal system of axes x′Ox and y′Oy, consider the points B  1 ; 4  and C  5 ;  4  .
1) Plot the points B and C.
2) Show that y = 2x + 6 is the equation of line (BC).
3) Let (d) be the line passing through B and perpendicular to (BC).
a. Determine the equation of the line (d).
b. Verify that the point A  3 ; 2  is a point on (d).
4) Let I be the midpoint of [AC].
Calculate the coordinates of I, then prove that (BI) is parallel to y′Oy.
5) Let D be the symmetric of B with respect to I.
Show that the coordinates of point D are  1 ;  6  .
6) Determine the nature of quadrilateral ABCD. Justify.
7) Let N be the translate of D under the translation with vector ⃗⃗⃗⃗⃗
BA.
a. Determine the coordinates of point N.
⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗
b. Complete:BA
BD = ⃗⃗⃗⃗⃗⃗⃗
……
42
***Multiple choice questions:
No//
Questions
1
The point A(3; 1) and the line (d) with equation
y = 2 are given in an orthonormal system of axes
x’Ox and y’Oy. The equation of the line (d’)
a
Answers
b
c
y = −x + 4
y=1
x=3
Parallel
Perpendicular
through A and perpendicular to (d) is…
2
In an orthonormal system, the two lines:
(D1 ): y = (2 − √5)x − 5 and
(D2 ): y = (2 + √5)x + 5 are…
3
Intersecting at
B(0; 5)
In an orthonormal system, the two lines with Intersecting at
equations: y = 2x + 3 and 2y + x = 1 are…
the point
(1; 5)
43
Perpendicular
Parallel
Translation and Vectors:
(2001 – 1st Session)
Remark: It is not required to copy again the figure to
the right.
CIGE is a square and the points B, H, F and D are the
midpoints of its sides.
Copy again and complete the following sentences:
1) The symmetric of triangle ABC with respect to A is
the triangle ------------⃗⃗⃗⃗⃗ is
2) The translate of D by the translation of vector CA
---------3) ⃗⃗⃗⃗⃗
DB + ------- = ⃗⃗⃗⃗⃗
DF
4) ⃗⃗⃗⃗⃗
AF + ⃗⃗⃗⃗⃗
AH = --------
C
B
D
.
H
E
F
G
I
A
(2005 – 1st Session)
It is not required to reproduce the opposite figure.
In this figure, ABCD is a parallelogram of center O and the points E, F, G and H are the midpoints of the
sides.
Reproduce and complete the following
phrases:
1) The symmetrical of triangle GOD about
point O is the triangle ---------2) The image (translate) of E by the
translation of vector ⃗⃗⃗⃗⃗
AO is the point ---3) The point F is the image (translate) of point
------- by the translation of vector ⃗⃗⃗⃗⃗
DO.
⃗⃗⃗⃗ .
⃗⃗⃗⃗ + --------- = FG
4) FE
5) ⃗⃗⃗⃗⃗
AE + ⃗⃗⃗⃗⃗
AH = -----------⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗
6) FE
BC = ------------
(2013 – 2nd Session)
⃗⃗⃗⃗⃗ = DA
⃗⃗⃗⃗⃗ and CF
⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗
Given a parallelogram ABCD with center O. The points E and F are such that: AE
OC.
1) Draw a figure.
2) Prove that ⃗⃗⃗⃗⃗
EB = ⃗⃗⃗⃗⃗
AC and ⃗⃗⃗⃗⃗
OF = ⃗⃗⃗⃗⃗
AC.
3) The lines (EF) and (OB) intersect at K. Prove that K is the midpoint of the segment [EF].
44
***Multiple choice questions:
No//
Questions
a
B is the
Answers
b
C is the
1
OAB is a triangle. The points C and F are so
midpoint of
midpoint of
Therefore:
[CF]
[BF]
ABCD is a parallelogram and E is the
C is the
translate of D under the translation with
midpoint of
that:
2
3
⃗⃗⃗⃗⃗
OA + ⃗⃗⃗⃗⃗
OB = ⃗⃗⃗⃗⃗
OC
and
⃗⃗⃗⃗⃗
AF = ⃗⃗⃗⃗⃗
OC.
vector ⃗⃗⃗⃗⃗
BA. Therefore…
[DE]
⃗⃗⃗⃗⃗ =
ABCD is a parallelogram, then ⃗⃗⃗⃗⃗
AB + DA
⃗⃗⃗⃗⃗
BC
45
c
F is the midpoint
of [BC]
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗ + DE
⃗⃗⃗⃗⃗ = DA
DB
⃗⃗⃗⃗⃗ = DB
⃗⃗⃗⃗⃗ + DA
⃗⃗⃗⃗⃗
DE
⃗⃗⃗⃗⃗
CA
⃗⃗⃗⃗⃗
DB
Trigonometry
(2001 – 2nd Session)
The unit of length is the centimeter. Consider a triangle ABC and designate by H the foot of height
2
relative to [BC], (H is between B and C). Given AC = 6 cm and cos HĈA =
3
1) Calculate CH and AH.
̂ A = 2√5. Calculate BH and BA.
2) Knowing that tan HB
5
3) Calculate the area of triangle ABC.
(2004 – 1st Session)
In this problem, the unit of length is 1 cm.
x and y are two positive numbers. ABC is a right triangle at A such that: AB = 2x + y,
3
and BC = 3x + y. The perimeter of triangle ABC is 24 and tan A𝐵̂C = .
4
 2x  y  0
1) Justify that the preceding given is translated into the system 
6x  3y  24
2) a- Calculate x and y showing all the steps followed.
b- Deduce the length of the sides of triangle ABC.
(2007 – 1st Session)
𝑥 is any acute angle, establish the following equalities:
1) (1 + tan2 x)cos 2 x = 1
2) (cos x + sin x)2 − 2 cos x sin x = 1
(2008 – 1st Session)
Consider an isosceles triangle ABC, such that AB = AC = 3 cm, and BC = 2 cm.
M is the midpoint of [BC].
1) a. Calculate AM.
̂C
b. Calculate sin AB
2) a. Calculate the area S of triangle ABC.
̂C
b. Show that 2S = BA × BC × sin AB
46
AC = x + y
(2013 – 1st Session)
𝛼 is an acute angle, Prove that: (1 − sin2 α)tan2 α = sin2 α
(2015 – 2nd Session)
1) a. Verify that x2 + 4x + 3 = (x + 2)2 – 1
b. Factorize x2 + 4x + 3.
2) Given an isosceles triangle ABC with vertex A so that its area is equal to x2 + 4x + 3 and BC = 2x + 2
(x > 0). Let [AH] be an altitude in this triangle.
a. Show that AH = x +3.
b. Calculate AB2 in terms of x.
3) a. Find x such that the area of ABC is equal to 8.
b. For x = 1, Calculate sin 𝐴𝐵̂ 𝐶 and deduce, rounded to the nearest degree, the measure of the angle
̂C
AB
(2019 – 1st Session)
3) Given P(x) =  2x  1  2x 2  9x  4 .
2
d. Verify that  2x  1 x  4   2x 2  9x  4 .
e. Show that P(x) =  2x  1 x  3 .
f. Solve the equation P(x) = 0.
Px
4) Let H(x) = 2
.
4x  1
d. Factorize 4x 2  1 .
e. For what values of x, is H(x) defined?
f. Simplify H(x).
5) Let ABC be a right triangle at A so that: AB = x − 3 and BC = 2x − 1 where x > 3.
a. Verify that sin BĈA = H(x).
b. Is there a value of x so that BĈA = 300 ? Justify.
47
***Multiple choice questions:
No//
1
Questions
1
𝑥 is an acute angle such that cos 𝑥 = 3 ,
5
2
OI = 2 − cos 600 =
3
A triangle ABC is right at B. If AB = 3 and AĈB =
400 , then AC =
2
𝑥 is the measure of an acute angle so that sin 𝑥 = 5
Then cos 𝑥 =
5
c
1
5
26
25
1
3
1
2
3
3
sin 400
3sin 400
sin 400
3
3
5
4
5
√21
5
1
2
√2
2
√3
2
−
Then sin 𝑥 =
4
a
Answers
b
ABC is a right triangle
̂ =y
at B such that BAC
̂ = 2y where y
and BCA
is a real number.
̂ is
The value of cos BAC
48
Statistical Survey
(2001 – 1st Session)
During a test, the grades of students of a class are given in the following statistical table:
Grades
8
10
12
Frequencies 3
6
5
1) What is the number of students of this class?
2) What is the mean grade of this class?
14
3
16
2
18
1
(2002 – 1st Session)
The opposite bar diagram represents a statistical series
1) Calculate the total frequency.
2) Represent this series in a table showing the frequencies, and
the relative frequencies in percentage.
3) Calculate the mean of this series.
(2003 – 2nd Session)
A statistical series is given in the opposite
table where a, b and c are integers and d
is a decimal number.
1) Calculate the numerical value of each of
the numbers a, b, c and d.
2) Calculate the mean of this series.
(2004 – 2nd Session)
The opposite circular diagram represents the distribution of marbles
in a bag according to their colours :
red : r, green : g , yellow : y , white : w , brown : b.
[EG] and [PH] are two diameters of the circle,
̂ E = 60° and DO
̂ H = 90°.
DO
̂ P , PO
̂ G and GO
̂ H.
1) Calculate the angles EO
2) Justify that the yellow colour is the most frequent.
3) Knowing that the number of the red marbles is 270. Reproduce
and complete the following table and verify that the number of
marbles in the bag is 1080 :
Colour
Frequency
r
270
g
4) Calculate the percentage of the red marbles.
49
y
w
b
(2005– 1st Session)
1) Solve the following system, showing all the steps of calculation :
x  y  11

2x  5y  34
2) A survey was made to find the number of books read by the students of a certain class. The results are
grouped in the following statistical table.
Number of read books
1
2
3
4
5
6
Number of students
5
x
4
3
y
2
We know moreover that the number of students of this class is 25 and the mean of read books is 3.
Calculate x and y.
(2006 – 2nd Session)
The opposite bar graph represents a statistical series.
1) Calculate the total frequency.
2) Represent this series in a table showing the frequencies,
and the relative frequencies in percentage.
3) Calculate the mean of this series.
(2007– 1st Session)
A statistical series is given in the
opposite table where a, b, c and d are
integers.
1) Calculate the numerical value of each
of the numbers a, b, c and d.
2) Calculate the mean of this statistical
series.
(2008 – 1st Session)
In what follows, we have the survey of the scores of 30 students of a class:
12; 18; 15; 11; 14; 7;14; 12; 11; 8; 15; 18
7; 18; 12; 14; 17; 10;14; 11; 10; 18; 17; 12
7; 12; 15; 8; 14; 17.
1) Use the above survey to construct a table of scores containing the frequencies and the increasing
cumulative frequencies.
2) What is the percentage of the students who got a score less than 13?
3) Calculate the mean of the class scores.
50
(2008 – 2nd Session)
The 300 students in a school are distributed into 5 categories
according to their ages, which are respectively 14, 15, 16, 17 and 18
years.
The circular diagram shows the percentage frequencies of this
distribution.
1) Copy and complete the following table:
2) Determine the mean age of these students.
3) Determine the number of students having ages strictly less than 17 years.
(2009 – 2nd Session)
The following table represents the grades over 60 of 20 students:
1) What is the relative frequency of the grade 52?
2) Complete the previous table.
3) Plot the increasing cumulative frequency polygon.
4) What is the percentage of students having a grade greater than 57?
5) Determine the average grade (mean) of the students.
(2010 – 1st Session)
200 people are surveyed about their favorite football team. The following table represents the result of
this survey.
1) Calculate a, b, c, d, e, f and g.
2) Draw a bar graph of frequencies.
3) Construct the corresponding circle graph.
51
(2011 – 1st Session)
The adjacent graphic represents the cumulative frequency polygon of the students' grades in a certain
class.
1) What is the number of the students of this class?
2) Complete the following table:
3) Write as percent the relative frequency of grade 10.
4) What is the average grade of the students of this class?
(2012 – 1st Session)
In what follows, are the scores of a student in five tests: 10; 8; 13; x and y.
The difference between x and y is 7. The average (mean) of these five scores is 12.
1) Write a system of two equations with two unknowns modeling the given situation.
2) Solve the obtained system.
(2013 – 1st Session)
40 students were surveyed about the number of books they read last month.
The following table represents the results of the survey:
1) Determine the mean of this series.
2) Copy the table above, then complete it.
3) What is the number of students who have read at least 6 books?
52
(2014 – 1st Session)
The following table shows the thicknesses, in cm, of 40 books:
1) Copy and complete the given table.
2) Calculate the average (mean) thickness of these 40 books.
3) Determine the percentage of books with a thickness between 2.75 and 3.75 cm.
(2015 – 2nd Session)
The Brevet students of five schools A, B, C and D sit for the official exam.
The adjacent circle graph represents the distribution of students in these schools.




The total number of students is 240.
The angle that represents the students of D and E together
is150⁰
The angle that represents the students of A is 90⁰
The number of students of B is equal to that of C.
1) Verify that the number of students of A is 60.
2) Calculate the number of students of B and that of C.
3) Show that the number of students of students D and E together is 100.
4) 20% of the students of A and 15% of the students of B failed.
Calculate the total number of students of A and B who passed the exam.
5) Three times the number of students of D minus the number of students of E is equal to 180.
a. Write a system of two equations with two variables to represent the number of students of D and E.
b. Solve the system and verify that the number of students of D is 70.
(2018 – 2nd Session)
Given 𝐴(𝑥) = 2𝑥 2 − 6𝑥 − (𝑥 − 3)(𝑥 − 1)
1) a. Show that 𝐴(𝑥) = (𝑥 + 1)(𝑥 − 3).
b. Solve the equation 𝐴(𝑥) = 0.
2) Verify that A(x) = x 2 − 2x − 3.
3) The grades of students, in Mathematics, are given in the following table. (x is a natural number)
Grades
Number
students
of
4
1
9
x2
12
x
Calculate x, knowing that the average (mean) of the grades is 10
53
19
1
Total
x +x+2
2
(2019 – 1st Session)
 x  y  16
4) Solve the following system: 
.
2x  3y  38
5) The following table represents the distribution of electronic games in a shop according to their prices:
4 000
5 000
6 000
Price of an electronic game (in LL) 3 000
9
m
15
n
Number of electronic games
a. The total price of all electronic games in this shop is 178 000 LL.
Show that this information is modeled by the following equation: 2m + 3n = 38.
b. Knowing that the total number of electronic games in this shop is 40. Calculate m and n.
6) Given: m = 10 and n = 6.
Calculate the average price (mean) of these 40 electronic games.
(2019 – 3rd Session)
The grades over 20 of 75 students, in Math, are given in the following table:
Grades
Number of students
8
12
10
18
11
30
x
15
1) Knowing that the average (mean) of the grades is 11.08, show that x = 15.
2) Complete the table by showing the increasing cumulative frequencies and the frequencies in
percentage.
3) Draw the corresponding circle graph (pie chart).
4) The principal confirmed that 75% of these students have a grade less than or equal to 11. Is he right?
Justify.
***Multiple choice question:
No//
1
Questions
a
Answers
b
c
13
14
14.5
The five grades of a student over 20 are:
10 ; 12 ; 13 ; 16 and 19.
The average grade is:
54
Thales – Similar triangles
(2003 – 1st Session)
ABCD is a rectangle such that AB = √7 + √3 and BC = √7 − √3.
1) Indicate its length and its width.
2) Calculate the exact value of AC and give an approximate value of AC to the nearest 10−2.
3) Let M be the point on [AB] such that AM = √7 . The line (DM) cuts the line (CB) in E.
Calculate EB.
(2007 – 2nd Session)
In the opposite figure (not drawn to scale):
3

ABC is a triangle right angled at A such that AB = 6 cm and tan AĈB = 2

MNP is a triangle similar to ABC such that:
MN MP 5
=
=
AB
AC 4
1) Find rounded to the nearest degree, the measure of
the angle AĈB and write on your paper the
measure of AĈB appearing on your calculator.
2) Prove that the triangle MNP is right angled at M
̂N
and that AĈB = MP
3) Calculate NP.
(2013 – 1st Session)
1) Solve the following system, showing the calculation details: {
2) In the opposite figure, the unit of length is the
centimeter:
The points O, K and P are collinear
The points O, B and A are collinear
(KB) and (PA) are parallel
OK = x, OP = y, KB=5, KP =6 and PA=8.
Calculate the length OP.
55
𝑥 − 𝑦 = −6
8𝑥 − 5𝑦 = 0
(2014 – 1st Session)
1) Rationalize the denominators of the following fractions
2) Consider a triangle ABC such that AB=
6
√7−1
; AC =
6
√7−1
6
√7+1
and
6
√7+1
and BC = 4.
a. Calculate AB2 and AC2. Deduce that the triangle ABC is right at A.
b. Let M be the midpoint of [BC] and E a point on the ray (semi-line) [AM) such that AE =
10
3
Calculate AM and ME.
3) Denote by F the orthogonal projection of C on (AE).
Show that the two triangles CAF and BCA are similar. Calculate CF.
***Multiple choice questions:
No//
1
Questions
a
Answers
b
c
AN
AM
AE
AF
AN
MN
(ME) and (NF) are two
parallel lines, then:
NF
ME
=
56
Geometry
(2001 – 1st Session)
Consider a circle (C) of center O, of radius R and of diameter [AB]. Designate by (d) the tangent at B to
circle (C). Let M be any point on (d). The perpendicular drawn from A to (AM) cuts (d) in N. The parallel
drawn from O to (AN) cuts [AM] in E and [BN] in F.
1) Draw the figure.
2) Prove that F is the midpoint of [BN].
3) a. Prove that the two triangles OEA and MBA are similar.
b. Prove that AE × AM = 2R2
4) Suppose, in this question, that M moves on the straight line (d) and designate by I the midpoint of
[OM]. Find the locus of I.
(2001 – 2nd Session)
Consider a circle (C) of center O and diameter [AB]. Let L be a point of this circle distinct of A and B.
̂ B cuts again (C) in P.
The bisector of the angle AK
1) Draw a figure.
̂ P.
2) Calculate the measure of angleAO
3) a. Construct point L, the transform of A by the translation which transforms O in P.
b. What is the nature of the quadrilateral OALP? Justify the answer.
(2002 – 1st Session)
Let (C) be a circle of center O and radius 3 cm. [AB] is a diameter of (C), E is a point of (C) such that
AE= 2 cm and M is a variable point of (C). Let I be the midpoint of [OB]. The straight line passing
through I and parallel to (OM) cuts (AM) in N.
1) Draw the figure.
2) a. Show that triangle AEB is right angled at E.
b. Calculate the exact value of BE.
̂ E to the nearest 10-2, and then deduce the measure of
c. Calculate the approximate value of tan BA
̂ E rounded to the nearest degree.
angle BA
3) Calculate IN.
(2002 – 2nd Session)
ABC is an isosceles triangle with principal vertex A, such that BC = 6 cm and AC = 5 cm.
The circle (O) with diameter [AC] and center O cuts the segment [BC] at F.
1) Draw a figure.
2) a. Prove that AFC is a right triangle.
b. Prove that F is the midpoint of [BC].
c. Calculate AF.
3) Let M be the image of A by the translation of vector ⃗⃗⃗⃗
FC .
a. What is the nature of quadrilateral AFCM? Justify your answer.
b. Are the points F, O and M collinear? Why?
c. Prove that M belongs to the circle (O).
57
(2003 – 1st Session)
Consider a circle (C) of diameter [AB], of center O and radius 4. Let G be a point of [OB].
The perpendicular to (AB) at G cuts the circle in two points M and N. The line (MO) cuts again the circle
(C) in point P.
1) Draw a figure.
2) Prove that the two triangles MGB and MAP are similar and deduce that: MA×MB = 8×MG.
3) Let E be the midpoint of [MA]. Prove that the two lines (OE) and (BM) are parallel.
(2003 – 2nd Session)
In the opposite figure:
 [AB] is a diameter of circle (C)
 OA = OB = 4
 OI = IE = 2
 (OI) is perpendicular to (AB).
̂ B = 900 .
1) Justify that AM
̂ I.
2) a. Calculate tan OA
̂ M.
b. Calculate, rounded to the nearest degree, the angle BA
c. Using sin 𝐵𝐴̂𝑀, calculate the length MB.
3) a. Calculate the exact value of AI.
b. Prove that the two triangles OAI and MAB are similar.
c. Calculate the exact value of the length MA.
4) Prove that the four points O, B, M and I belong to the same circle whose center is to be determined.
(2004 – 1st Session)
In the opposite figure, we have:
 AB = 8 cm
 (C) is the circle of diameter [AB] and center O.
 M is the point of segment [AO] such that AM = 3cm.
 ( C ) is the circle of diameter [AM]
 D is a point of (C) such that BD = 7 cm.
 (C) and ( C ) are tangent at A.
1) Reproduce the figure.
2) Justify that ADB is a right triangle, and calculate,
̂ D.
rounded to the nearest degree, the measure of angle AB
3) The straight line (AD) cuts the circle (C′) in a second point E. Prove that (BD) and (ME) are parallel,
then calculate EM.
4) The common tangent at A to (C) and (C′) cuts the straight line (BD) in N. Choose two triangles and
prove that they are similar, then deduce that AN2 = ND × NB.
5) Let F be the point such that ⃗⃗⃗⃗⃗
DF = ⃗⃗⃗⃗⃗
DA + ⃗⃗⃗⃗⃗
DB.
Prove that the quadrilateral DAFB is a rectangle. Deduce that F belongs to circle (C).
58
(2004 – 2nd Session)
ABC is a triangle right angled at A such that: AB = 4 cm and AC = 6 cm. M is the midpoint of [AC].
1) Calculate BM.
̂ M and deduce the angle BM
̂ C.
2) Calculate, rounded to the nearest degree, the angle AB
3) Let E be the symmetric of B with respect to M.
a. Place E and determine the nature of quadrilateral CBAE.
b. Calculate the area of CBAE.
4) Let G be the symmetric of B with respect to the straight line (AC).
a. Place G and determine the nature of quadrilateral GECA.
b. Calculate the area of quadrilateral BCEG.
(2004 – 2nd Session)
Two perpendicular straight lines (d) and (d′) intersect in a point O. The circle (C) of center O and radius 4
cuts (d) in A and B. Let M be a point of (C) distinct from A and let L be the midpoint of [AM]. The line
(AM) cuts (d′) in N.
1) Draw a figure.
2) What is the nature of triangle OLA? Justify.
3) a. Prove that the two triangles OAN and MAB are similar.
b. Deduce that the product AM × AN remains constant when M moves on circle (C).
(2005 – 1st Session)
In the opposite figure:
 (C) is a circle of center O, [AB] is a fixed diameter of (C)
such that AB = 6 cm
 [MN] is a variable diameter of (C).
 E is the symmetrical of A with respect to M.
1) Reproduce this figure.
2) a. Prove that (OM) and (BE) are parallel.
b. Prove that (BM) is the perpendicular bisector of [AE].
c. Prove that triangle ABE is isosceles of principal vertex B
3) Let I be the intersection point of the straight lines (EN) and (AB).
a. Prove that the two triangles ION and IBE are similar and
deduce that: IB = 2 × IO.
b. Calculate IO and IB.
c. Is I the center of gravity of triangle MBN? Justify.
d. (EN) cuts (MB) in F. Prove that (OF) is perpendicular to (MB).
(2005 – 2nd Session)
EBF is a triangle right angled at B such that EB = 6 cm, BF = 8 cm and FE = 10 cm. M is the midpoint of
[BF] and (C) is the circle of diameter [MF]. The circle (C) cuts again [EF] in G.
The straight lines (MG) and (EB) intersect in S.
1) Draw a figure.
2) Prove that the four points E,B, M and G belong to the same circle whose diameter is to be determined.
59
3) a. Prove that the two triangles EBF and MGF are similar and calculate MG and GF.
b. Calculate the area of triangle MGF.
c. Calculate the ratio of the areas of the two triangles EBF and MGF.
4) Let P be the point of intersection of (EM) with (SF).
a. Prove that (EP) is perpendicular to (SF).
b. Deduce that P is a point of circle (C).
(2006 – 1st Session)
In the opposite figure:
 (C) is a circle of center O and diameter [AB]
 OA = OB = 3 cm
 P is the point of [AB) such that OP = 5 cm
 E is a point of (C) such that PE = 4 cm
 (D) is the tangent at A to (C)
 M is a variable point on (D)
 (PE) cuts (D) in J.
1) Reproduce this figure. It will be used and completed by the
remaining parts of this problem.
2) a. Prove that (PE) is tangent to (C) at E. Deduce that JE = JA.
̂E and round to the nearest degree the angle OP
̂E.
b. Calculate tan OP
3) Let JE = JA = x and JP = x + 4 where x is a measure of length in centimeters.
a. Apply Pythagoras theorem to triangle APJ and calculate x.
b. Deduce that triangle ABJ is a right isosceles triangle.
4) (JB) cuts (C) in a second point F. Prove that F is the midpoint of [JB] and that (FO) is the
perpendicular bisector of [AB].
(2006 – 2nd Session)
Consider a semi-circle (C) of diameter [AB], of center O and radius R. Let E be the midpoint of segment
of segment [OB]. The perpendicular bisector of [OB] cuts (C) in G. Let K be a variable point on segment
[EG]. The straight line (BK) cuts (C) in a second point M.
1) Draw a figure.
̂ G.
2) Prove that OB = OG = GB. Deduce the measure of the angle BO
3) Calculate, in terms of R, the area of the triangle AGB.
4) a. Prove that the two triangles BEK and BMA are similar.
b. Deduce that BK × BM = BA × BE.
60
(2007 – 1st Session)
Consider a circle (C) of center O and diameter [AB] such that AB = 6cm. E is a variable point of (C) and
M is symmetric of A with respect to E.
The straight line (BM) cuts circle (C) in a second point P.
Designate by J the point of intersection of (BE) with (AP), T the
point of intersection of (AB) with (MJ) and S the midpoint of [MB].
1) Draw a figure.
2) Prove that triangle ABE is right.
3) Prove that triangle ABM is isosceles of principal vertex B.
4) Prove that triangle ABM is an enlargement of triangle OBS and
precise the scale factor of this enlargement.
5) a. Prove that (AT) is perpendicular to (MJ).
b. Prove that the points E, B, T and M belong to the same circle. Determine a diameter of this circle.
(2007 – 2nd Session)
Consider a semi-circle (C) of center O, radius R and diameter [AB]. Let M be a point on (C) distinct from
A and B. The tangent at M to (C) cuts the tangent at A in point N and the tangent at B in point P. (OP)
cuts [MB] in D and (ON) cuts [AM] in E.
1) Draw a figure.
2) Prove that D is the midpoint of [MB] and that E is the midpoint of [MA].
3) Calculate ED in terms of R.
4) Prove that ODME is a rectangle.
(2008 – 1st Session)
In the figure below we have the 2 circles (C1) and (C2), of the
same center O and their radii are R1 = 2cm and R2 = 4 cm
respectively.
A straight line passing through O cuts (C1) at F and A, and cuts
(C2) at B and E. The tangent to circle (C1) at A cuts the circle
(C2) at C and D.
1) Show that (CD) is the perpendicular bisector of [OB].
2) Determine the nature of the triangle OBC.
3) Show that the quadrilateral OCBD is a rhombus.
4) Let P be the midpoint of [CE].
a. Calculate OP and deduce that P belongs to circle (C1).
b. Show that D, O and P are collinear.
c. Show that (OP) is perpendicular to (CE) and deduce that (CE) is tangent to (C1).
d. Show that (CO) is perpendicular to (DE).
5) Show that the triangle EBC is an enlargement of the triangle EOP and precise the center and ratio of
this enlargement.
61
(2008 – 2nd Session)
Consider a circle (C) with center O, diameter [AB] and radius 3 cm.
(d) is the tangent at A to (C) and F is a point on (d) such that AF = 4 cm.
E is the orthogonal projection of A on (OF); (AE) cuts again (C) at a point L.
1) a. Draw a figure.
b. Calculate OF and cos OF̂A.
2) a. Show that the two triangles OAF and BLA are similar. Write the similarity ratio.
b. Use this ratio to calculate BL and AL. Deduce OE and AE.
3) (LO) cuts again (C) at K, and (BK) meets (d) at S.
a. Determine the nature of quadrilateral BALK.
̂ K and AB
̂K
b. Compare SA
̂ K in the triangle SAK, and then calculate AS.
c. Determine cos SA
̂ in each of the two right triangles ABK and ABS, then deduce the
d. Express the cosine of the angle B
2
relation AB = BK× BS.
(2009 – 1st Session)
Consider a circle (C) with center O, diameter [AB] and radius 2 cm. T is a point on (C) so that AT = 2 cm,
and M is the symmetric of O with respect to A.
1) a. Make a figure.
b. Prove that (MT) is tangent to (C).
c. Calculate MT.
d. Prove that MTB is an isosceles triangle.
2) E is the meeting point of (MT) and the tangent at B to (C).
a. Prove that T is the midpoint of [EM].
b. (TO) intersects (C) at F, calculate EF.
c. Calculate to the nearest degree the angle EF̂T .
⃗⃗⃗⃗⃗⃗ .
3) N is a variable point on (C) and S is the image of N under the translation with vector AM
Prove that ASNO is a rhombus.
(2009 – 2nd Session)
In the opposite figure:
 (S) is a semicircle with center O and radius R
 [EF] is the diameter of (S)
 A is a point on (EF) so that OA = 2 R
 (d) is a variable line through A that intersects (S) at B and C
 The tangents at B and C to (S) intersect at M.
1) Justify that (OM) is the perpendicular bisector of [BC].
2) (OM) intersects [BC] at I. Let P be the orthogonal
projection of M on (OA).
a. Show that the triangles OIA and OMP are similar.
b. Prove the relation OA × OP = OM × OI.
62
3) a. Let (S′) be the circle circumscribed about the triangle CIM. Show that (OC) is tangent to (S′).
b. Use two similar triangles to prove that OM × OI = R2.
c. Calculate OP in terms of R.
4) In this questions, suppose that OBMC is a square.
a. Calculate the lengths BC and MP in terms of R.
̂P .
b. Calculate the exact value of tan MA
̂ P to the nearest degree.
c. Calculate the value of MA
(2010 – 1st Session)
In the adjoining figure:
 (C) is a circle with center O and radius R.
 [AB] is a diameter of (C)
 M is a variable point on (C)
 AMNB is a parallelogram.
1) Copy this figure which will be completed in the other parts of the
problem.
2) Let E be the symmetric of N with respect to B. Prove that
AMBE is a rectangle.
3) (NO) intersects [MB] at G.
a. Prove that (EG) intersects [MN] at its midpoint.
b. Prove that the two triangles GOB and GNM are similar, then calculate
GN
GO
4) O' is the point defined so that ⃗⃗⃗⃗⃗⃗
BO′ = ⃗⃗⃗⃗⃗
OB. Show that MOO'N is a parallelogram.
(2010 – 2nd Session)
In the following figure where the unit is the centimeter:
 ABCD is a right trapezoid
 AB = 3; AD = 4; CD = 5
 The lines (AB) and (CD) are parallel
 The lines (AC) and (BD) intersect at O
1) Reproduce the figure.
2) Show that the triangle BCD is isosceles with principal vertex D.
3) Calculate the area of the trapezoid ABCD.
4) Show that OA× OD = OC × OB.
5) The lines (AD) and (BC) intersect at S. Show that the angles 𝐶𝐵̂ 𝐷 and 𝐴𝐵̂ 𝑆 are equal.
6) Suppose that SA = x.
x
3
a. Show that x+4 = 5 :, then calculate SA.
b. Determine the value of the angle AŜB rounded to the nearest degree.
c. Let H be the midpoint of [BC]. Show that the four points A, B, H and D are on the same circle
whose diameter to be determined.
63
(2011 – 1st Session)
̂ E = 1400 .
ABE is an isosceles triangle with vertex B, and so that BE = BA = 6 cm and AB
The circle (C) with diameter [BE] and center O intersects (AB) at another point F.
1) Make a figure.
2) What is the nature of the triangle BEF? Justify.
3) I is the midpoint of [AE]. Show that I is a point on (C).
̂ E and EB
̂F .
4) a. Calculate BA
b. Find to the nearest thousandth an approximate value of BF.
5) Prove that the two triangle ABI and AEF are similar, and deduce that AB × AF = 2 × AI2.
6) G is the translate of E under the translation with vector ⃗⃗⃗⃗
FB.
a. Show that EFBG is a rectangle but not a square.
b. Prove that G, O and F are collinear.
(2011 – 2nd Session)
In the following figure:
 (C) is a circle with diameter [AB], center O and radius 3 cm
 The perpendicular at O to (AB) intersects (C) at E and J
 The bisector of the angle 𝐸𝐴̂𝐵 intersects [OE] at H and
intersects (C) at another point G.
 The line (BG) intersects (AE) at K and (OE) at F.
1) Reproduce the figure.
0
̂ G = 45
2) Verify that BA
2
3) Prove that the triangle ABK is isosceles with vertex A.
4) Calculate AE and EK.
5) Prove that the two triangles AOH and AGB are similar. Deduce
the value of the product AH × AG.
450
6) a. Using cos ( 2 ) in the triangle AOH, calculate AH to the
nearest hundredth.
b. Deduce an approximate value of the similarity ratio of the triangles AOH and AGB.
7) (BH) and (AF) intersect at I, prove that I is a point on (C).
(2012 – 1st Session)
Consider a semicircle (C) with center O and radius R. [AB] is a diameter of (C) and D is a point on (C) so
that BD=R.
Let M be the midpoint of [OA]. The perpendicular bisector of
[OA] intersects [AD] at E, (BD) at F and (C) at K.
1) Reproduce and complete the figure.
2) Calculate the angles of the triangle ABD, then calculate AD in
terms of R.
3) a. Prove that the two triangles ADB and FMB are similar.
b. Calculate BF in terms of R.
64
4) (BE) intersects (AF) at J. Show that J is on (C).
5) Prove that the points B, M, J and F are on the same circle whose center and radius should be
determined.
6) a. Show that the triangle OKA is equilateral.
b. Prove that (OK) passes through the midpoint of [AF].
(2012 – 2nd Session)
In the adjoining figure:
 A and B are two fixed points
 (d) is the perpendicular at A to (AB)
 C is a variable point (d)
 M is the midpoint of [AC]
 E is the symmetric of B with respect to M
 (L) is the circle with diameter [AE] and center I.
1) Reproduce this figure.
2) Prove that the quadrilateral ABCE is a parallelogram.
3) Let F be the translate of B under the translation with
vector ⃗⃗⃗⃗⃗
CA . Show that E, A and F are collinear.
4) (L) intersects (AB) at a a second point G.
a. Prove that ACEG is a rectangle. Deduce that G is the translate of A under the translation with
vector ⃗⃗⃗⃗⃗
BA.
b. Prove that the two triangles AGM and BGF are similar.
(2013 – 1st Session)
In the opposite figure:
 A, E and B are three collinear points
 AE = 8 and EB = 4
 (C1) is the circle with diameter [EB] and center J
 (C2) is the circle with diameter [EA] and center I
 M is a variable point on (C1)
 The line (ME) intersects (C2) at another point N.
1) Copy this figure.
2) Show that the lines (MB) and (NA) are parallel.
3) Prove that the triangle ANE and BME are similar. Determine the ratio of similarity.
⃗⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗⃗
4) Let P be the point defined as ⃗⃗⃗⃗⃗⃗
MP = ME
MB.
a. Prove that the quadrilateral EPBM is a rectangle.
b. Deduce that P is a point on (C1).
5) Denote by K the intersection point of (ME) and (IP). Prove that [MK] is a median in the triangle IMP.
65
(2013 – 2nd Session)
In the next figure:
 (d) and (d') are two lines intersecting at D
 EDC is a right triangle at E
 AD = 6 cm and DE = EC = 2 cm
 (AH) is perpendicular to (d')
1) Copy this figure.
2) What is the nature of the triangle ADH?
Calculate the exact lengths of [DH], [DC] and [AC].
3) Prove that the points A, H, E and C are on the same circle whose diameter to be determined.
4) Let M be the orthogonal projection of D on (AC).
Prove that the two triangles AHC and DMC are similar. Calculate the product CM × CA.
5) Calculate tan AĈH . Deduce the value of the angle AĈH rounded to the nearest degree.
6) The lines (AH) and (CE) intersect at B; prove that (MD) passes through B.
(2014 – 1st Session)
In the figure to the right:
 (C) is a semicircle with center O and radius 5 cm,
 [BD] is the diameter of (C),
 A is a point on (C) such that AB = 6 cm.
1) a. What is the nature of triangle ABD? Justify.
b. Calculate AD.
2) The perpendicular through A to [BD] intersects it at H. Prove that the triangles AHB and DAB are
similar, and deduce that AH = 4.8cm.
3) The tangent at D to (C) intersects (BA) at E.
̂ then calculate DE.
a. In the two triangles ABD and BDE, write the ratios equal to tan B
̂ rounded to the nearest degree.
b. Determine the measure of B
4) Let P be the point on [BD] such that DP = 4 cm. The parallel through P to (AB) intersects [DE] at Q.
Calculate DQ.
5) Denote by R the translate of P under the translation with vector ⃗⃗⃗⃗⃗
QE.
a. Show that R is a point on [BE].
b. The line (PR) intersects [AD] at J. Prove that (BJ) is perpendicular to (DR).
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(2015 – 1st Session)
In the adjacent figure:
 (C) is a circle with center O and diameter AB = 6 cm.
 D be a point on (C) such that BD = 3.6 cm.
 M is the midpoint of [OB].
 The parallel through M to (BD) intersects [AD] at J.
1) Copy the figure; it will be completed in the following parts.
2) Show that ABD is a right triangle, and then verify that AD = 4.8 cm.
3) Verify that AJ = 3.6 cm and calculate JM.
4) The tangent to (C) at A and D intersect at L. The two lines (AD) and
(LO) intersect at F.
a. Calculate OF.
b. Prove that the two triangles OFA and OAL are similar, then calculate AL.
c. Calculate, rounded to the nearest degree, the measure of the angle AL̂D.
(2015 – 2nd Session)
In the adjacent figure:
 (C) is a circle with center O and diameter [IA] so that IA = 8 cm.
 B is a point on the tangent at I to (C) so that IB = 6 cm.
1) Copy the figure that will be completed later.
2) Let (C’) be the circle with diameter [IB].
The two circles (C) and (C’) intersect at I and another point E.
a. Prove that A, E and B are collinear.
b. Calculate AB.
̂A
3) a. Write in two different triangles the ratios equal to cos IB
b. Show that BE = 3.6.
c. Deduce the length AE, then calculate IE.
4) The tangent at B to (C’) intersects (IE) at F.
a. Show that the two triangles EBF and EIB are similar.
b. Deduce the value of EI x EF.
5) Let L be the translate of B under the translation with vector ⃗⃗⃗
IA.
Prove that the four points A, E, F and L are on the same circle with diameter to be determined.
(2016 – 1st Session)
In the adjacent figure:
 AE = 4 cm.
 (C) is the circle with diameter [AE] and center O.
 B is the symmetric of E with respect to A.
 (BD) is tangent to (C) at D.
1) Copy the figure.
2) Calculate BD.
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3) The parallel through point A to (OD) intersects the line (BD) at M and (ED) at L.
a. Show that D is the midpoint of [EL].
b. Deduce that M is the centroid of triangle EBL.
4) a. Prove that the two triangles BDE and BAD are similar.
b. Calculate
DE
DA
5) Let F be the translate of A by the translation with vector ⃗⃗⃗⃗⃗
ED
a. Prove that ADLF is a rectangle.
b. Prove that F is the midpoint of [BL].
c. Deduce that the points E, M and F are collinear.
(2016 – 2nd Session)
In the adjacent figure:
 (C) is a circle with center O and radius 5 cm
 [AB] is a diameter of this circle
 (d) is the tangent at B to (C)
 L is a point on (d) such that BL = 7.5 cm
 M is a point on [AB] such that AM = 4 cm.
1) Copy the figure.
2) Calculate the length AL.
̂ L.
3) Calculate cos BA
4) The line (AL) intersects the circle at E.
a. Prove that the two triangle ABL and BEL are similar.
Write the ratio of similarity.
b. Deduce that EL = 4.5 cm.
5) The perpendicular through M to (AL) intersects [AL] at
N and (d) at G.
̂ L in triangle MAN to verify that AN = 3.2 cm.
a. Use cos BA
b. Prove that
EB
NG
=
15
31
(2017 – 1st Session)
In the adjacent figure:
 (𝐷) and (𝐷’) are two perpendicular lines at A.
 O is a point on (𝐷) so that OA = 6.
 (C) is a circle with center O and radius 4.
 M is a point on the line (𝐷’) so that AM = 3.
 (MB) is a tangent through M to the circle (C).
 [BF] is an altitude in the triangle OBM.
1) Copy the figure.
2) Show that OM = 3√5.
3) a. Show that the triangles OFB and OBM are similar.
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b. Deduce that OF × OM = 16.
c. Calculate OF.
4) The two segments [BF] and [OA] intersect at I.
̂ A.
a. Write in the two triangles FOI and MOA the ratios equal to cos MO
b. Deduce that OI × OA = 16.
c. Calculate OI.
5) The line (FB) intersects (𝐷’) at E. Show that (MI) is perpendicular to (OE).
(2017 – 2nd Session)
In the adjacent figure:
 (C) is a circle with center O, radius 5 and diameter [EB].
 A is a point on (C) so that AE = 6.
 (d) is the tangent at B to (C).
 [AL] is an altitude in the triangle ABE.
1) Copy the figure that will be completed in the remaining
parts of the problem.
2) a. Calculate AB.
̂ B = 4.
b. Verify that sin AE
5
3) The parallel through L to (AB) intersects [EA] at M and the
line (d) at F.
a. Prove that the two triangles EML and FBL are similar.
b. Calculate, rounded to the nearest degree, the measure of angle BF̂L.
4) Prove that the points E, M, B and F are on the same circle whose center I should be determined.
5) Prove that the quadrilateral ALFB is a parallelogram.
6) The diagonals [AF] and [BL] of the parallelogram ALFB intersect at J.
Prove that (IJ) is perpendicular to (AB).
(2018 –1st Session)
In the adjacent figure:
 (C) is a semicircle of diameter [AB], with center O and
radius 6 cm;
 The perpendicular bisector of [AB] intersects (C) at D;
 E is a point on segment [OD] so that OE 4 cm;
 (AE) intersects (C) at F.
1) Reproduce the figure.
2) Verify that AE = 2√13 cm.
3) a. Prove that AFB is a right triangle at F.
b. Prove that the two triangles AOE and AFB are similar.
c. Deduce the value of AEAF.
69
4) The line (BF) intersects line (OD) at K and the line (BE) intersects line (AK) at I.
a. Prove that line (BE) is perpendicular to line (AK).
b. Deduce that I is a point on (C).
5) The tangent to (C) at A intersects (BE) at S.
a. Show that E is the midpoint of [BS].
b. Verify that BS = 4√13 cm.
(2018 – 2nd Session)
In the adjacent figure:



(C) is a semicircle with center O, diameter [AB] and radius 2 cm.
F is a point on (C) so that BF = 2 cm.
E is the symmetric of O with respect to B.
1) Reproduce the figure.
2) Verify that AF = 2√3 cm.
3) Show that (EF) is tangent to (C).
4) Let L be the midpoint of [OB]. Show that (FL) is
perpendicular to (OB).
⃗⃗⃗⃗ .
5) T is the point so that ⃗⃗⃗⃗
FT = LE
The parallel through T to (OF) intersects [EF] at R and [LE] at G.
a. Show that (TG) is perpendicular to (EF).
b. Show that the two triangles FLE and GRE are similar.
c. Deduce that
EG
ER
=
2√3
.
3
(2019 –1st Session)
In the adjacent figure:




ABCD is a square of side 4
M is the midpoint of [BC]
(AM) intersects (DC) at N
(d) is the la perpendicular through A to (AM).
1) Reproduce the figure.
2) Calculate AM.
NC
3) Calculate the ratio
, deduce that C is the midpoint of [DN].
ND
4) Lines (d) and (CD) intersect at Q.
70
̂ D = NA
̂ D.
a. Show that AQ
b. Show that the two triangles DAQ and DNA are similar.
Deduce that DQ × DN = 16.
c. Calculate DQ.
5) Show that the triangle AQM is a right isosceles triangle at A.
6) Let (C) be the circle with diameter [AQ] and L is the translate of Q under the translation with
⃗⃗⃗⃗⃗⃗ .
vector AM
Show that (LQ) is tangent to the circle (C).
(2019 – 2nd Session)
In the adjacent figure:



OABD is a rectangle so that OA = 5 and AB = 3
(C) is the circle of center O passing through point A
The line (BD) intersects the circle (C) at M and N.
1) Draw the figure.
2) a- What is the nature of triangle ONA? Justify.
̂ O.
b- Show that [NA) is the bisector of angle BN
3) Show that DN = 4 and calculate BN.
4) The two lines (NA) and (OD) intersect at L.
a- Show that two triangles BAN and OLA are similar.
b- Deduce that BN × LO = 15, then calculate LO.
5) The perpendicular through A to (OB) intersects circle (C) at F.
Show that (BF) is tangent to the circle (C).
(2019 – 3rd Session)
In the adjacent figure:





(C) is the circle with center O and diameter BE = 6
(C′) is the circle with diameter AB = 4
(C) and (C′) are tangent externally at B
[Ax) is tangent to (C) at F.
[Ax) intersects (C′) at M.
71
1) Draw the figure.
2) Calculate AF.
3) a. Calculate the ratio
BM
then deduce BM.
OF
8 10
.
7
4) Let (d) be the tangent to (C) at E, (d) intersects [Ax) at N.
Prove that the two triangles AMB and AEN are similar then deduce AN.
̂ M. Deduce, rounded to the nearest degree, the measure of angle 𝐴𝑁
̂ 𝐸.
5) Calculate cos AB
6) The line (FO) intersects (d) at I, and line (ON) intersects (AI) at K.
a. Prove that line (KN) is perpendicular to line (AI).
b. Deduce that triangle NAI is isosceles of vertex N.
̂ I.
7) Calculate the measure of angle NA
b. Verify that AM =
***Multiple choice questions:
No//
Questions
a
Answers
b
c
(√26 + √10)cm
(√√26 + √10)cm
6 cm
12
6√3
9
1
In
the
adjacent
figure, the area of
the square ABEF is
26 cm2 and the area
of the square ACHG
is 10 cm2.
Then BC = ……
2
A triangle ABC is right at A and M is the
midpoint of [BC].
If AM = AB = 6 then AC = ……
72