MEI Conference 2009
Stretching students:
AS Core
Presenter: Bernard Murphy
bernard.murphy@mei.org.uk
© MEI 2009
Workshop D1
1. Illustrating the root of a quadratic using circles (1)
Choose two positive numbers a and b
Draw perpendicular line segments PQ and
PR as shown along with the circle centre R
passing through P.
Prove that the length of QS is a root of the
equation x 2 = ax + b 2 .
-----------------------------2. Illustrating the root of a quadratic using circles (2)
Choose a quadratic of the form
y = x 2 − ax + b
Draw the line segment with endpoints
( 0,1) and ( a, b ) and construct the
circle with this line segment as
diameter.
Prove that this circle crosses the
x − axis at the roots of the equation
x 2 − ax + b = 0 .
-----------------------------3. The ladder problem
A ladder of length 15 feet leans against a cube of
side length 4 feet. Both the ladder and the cube
are in contact with the floor and the wall.
The wall and the floor are perpendicular.
Find the exact height to which the ladder will reach.
15
Hint: Although you might not be able to solve a
quartic equation like x 4 + 4 x3 − 6 x 2 + 12 x + 9 = 0
you can solve ( x 2 + 2 x + 3) − 16 x 2 = 0 .
2
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4
4. Isosceles Triangles and Surds
Both of the isosceles triangles in the diagram have
• all three vertices on grid points
• area 20 square units
• equal sides meeting at the origin.
In total, how many isosceles triangles
have these three properties?
-----------------------------5. Take a sheet of A4…
…and make two 45D folds as shown:
If the short edge of the sheet
has length 1 unit, what is the
perimeter of the new shape?
1
-----------------------------6. Splitting equilateral triangles
Prove that triangle with sides 3, 7, 8
contains a 60° angle.
Investigate triangles with sides
a = m 2 − n 2 , b = 2mn − n 2 , c = m 2 − mn + n 2
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7. AP over GP
How would you find the sum of an infinite number of
fractions where the numerators form an AP and the
denominators a GP?
1 2 3 4
For example, + + + + ...
2 4 8 16
or more generally
.
.
.
…
a a + d a + 2d a + 3d a + 4d
+
+
+
+
+ ...
A
Ar
Ar 2
Ar 3
Ar 4
1
1
1
2
1
1
4
1
4
1
16
…
8
1
16
…
8
1
16
…
8
1
16
…
-----------------------------8. A proof that there is an infinite number of primes
Assuming that there is only a finite number of primes...
⎞
1 1 1 1 1 1
⎛ 1 1
⎞⎛ 1 1
⎞ ⎛ 1 1
⎜1 + + 2 + +... ⎟ ⎜1 + + 2 + ... ⎟ ... ⎜ 1 + + 2 + ... ⎟ = 1 + + + + + + + ...
2 3 4 5 6 7
p p
⎝ 2 2
⎠⎝ 3 3
⎠ ⎝
⎠
How does this lead to a contradiction?
-----------------------------9. Powers of 9
The triangular numbers are formed by adding
consecutive numbers staring from 1.
The first few triangular numbers are
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105 …
●
●●
●●●
●●●●
Notice what happens if you add together the first few powers of 9:
1
1 + 9 = 10
1 + 9 + 92 = 91
Prove that the sum is always a triangular number.
© MEI 2009
1 + 9 + 92 + 93 = 820
10. The first digit in powers of 2
The table shows the first twenty powers of 2.
Of these twenty numbers, six have 1 as the first digit.
30% seems like a high proportion but in fact, as you look at
more and more powers of 2, the proportion with 1 as first digit
approaches log10 2 ≈ 0.30103 .
Here is how you prove this surprising fact.
1. Explain why there will be exactly one power of 2 with first
digit 1 for any given number of digits.
2. Explain why the number of digits in the number N is the
integer part of 1 + log10 N
1 + log10 2n
3. Explain the relevance of the fraction
and show
n
that this approaches log10 2 as n increases.
-----------------------------11. A lower bound for cos θ
Using the diagram of a sector of a circle,
prove that cos θ ≥ 1 −
θ2
2
for 0 ≤ θ ≤
π
2
-----------------------------12. Sum of arctangents
Explain how this diagram proves
1
1
arctan + arctan = 45D .
2
3
Find a diagram that proves
arctan1 + arctan 2 + arctan 3 = 180D
© MEI 2009
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2
4
8
16
32
64
128
256
512
1024
2048
4096
8192
16384
32768
65536
131072
262144
524288
1048576
13. Pythagoras and Tetrahedra
The diagram shows a right-angled tetrahedron
formed by slicing a corner off a cube.
If ΔOBC represents the area of triangle OBC,
prove that
2
2
2
2
( ΔOAB ) + ( ΔOBC ) + ( ΔOCA) = ( ΔABC )
-----------------------------14. Common tangent
A tangent touches the curve y = x 2 ( x − 1)( x − 3) at two distinct points. Find the
equation of this tangent.
-----------------------------15. Tangent to a cubic
M is the midpoint of BC. Prove that for any cubic with three real roots, the
tangent at D passes through A.
© MEI 2009
1. A function f : \ → \ has the property: f ( m + n ) = f ( m ) × f ( n ) for all m and n .
If f ( 4 ) = 16 what other values of f ( m ) can be determined?
m
f (m)
-4
-3
-2
-1
0
1
2
3
4
16
5
6
7
8
2. Prove that if a, b and c are distinct real numbers then the quadratic
equation ( x − a )( x − b ) + ( x − b )( x − c ) + ( x − c )( x − a ) = 0 has real roots.
3. The news on September 28, 2008 reported that a new prime number had
been discovered which had 12,978,189 digits. It is of the form 2 p − 1 where p
is a prime number. How might you find the value of p ?
4. (a) Use the facts that 32 > 23 and 35 < 28 to find upper and lower bounds for
the value of log 2 3 .
(b) Prove that log 2 3 is irrational
1 1 1 1
5. Select numbers from the sequence 1, , 2 , 3 , 4 ,... to form an infinite
2 2 2 2
1
geometric series with sum
where m is an integer. For which integers, m ,
m
is this possible?
6. What are the last two digits (i.e. the tens and units digits) in 3176 ?
x
7. Draw the graph of y = x .
8. Find three positive whole numbers a, b and c satisfying a 3 + b 4 = c 5 .
9. Evaluate
1
1
1
1
1
+
+
+
+ ... +
1+ 2
2+ 3
3+ 4
4+ 5
99 + 100
10. It just so happens that
2 23 = 2
2
3
. How many other equations are there
like this one?
11. Think about base 2 numbers beyond the “binary point”:
What is the base 2 number 0.1111.... ? What is
1
in base 2?
3
12. To test if a number is divisible by 11 you can check if the difference
between the sum of the digits in the even positions and the sum of the digits
in the odd positions is divisible by 11. For example, 3859276 is divisible by 11
if and only if ( 3 + 5 + 2 + 6 ) − ( 8 + 9 + 7 ) is divisible by 11. Why does this rule
work?
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