Intercepts:
1. What is the y-intercept of the line that passes through A(−1, 5) and B(1, 7)? 2020 1(b)
2. The lines with equations y = 3x + 7, y = x + 9, and y = mx + 17 intersect at a single point. Determine the
value of m. 2020 1(c)
Proportions:
1. Joyce has two identical jars. The first jar is 3 4 full of water and contains 300 mL of water. The second
jar is 1 4 full of water. How much water, in mL, does the second jar contain? 2019 1a
Digits / Integers:
1. The three-digit positive integer m is odd and has three distinct digits. If the hundreds digit of m equals
the product of the tens digit and ones (units) digit of m, what is m? 2020 2(a)
2. Suppose that n is a positive integer and that the value of n 2 + n + 15 n is an integer. Determine all
possible values of n. 2020 2(c)
3. Determine all integers k, with k 6= 0, for which the parabola with equation y = kx2 + 6x + k has two
distinct x-intercepts. 2020 3(c)
4. The positive integers a and b have no common divisor larger than 1. If the difference between b and a
is 15 and 5 9 < a b < 4 7 , what is the value of a b ? 2020 4(a)
5. What integer a satisfies 3 < 24 a < 4? 2019 1b
6. Determine the two pairs of positive integers (a, b) with a < b that satisfy the equation √ a + √ b = √ 50.
2019 5a
7. A five-digit integer is made using each of the digits 1, 3, 5, 7, 9. The integer is greater than 80 000 and
less than 92 000. The units (ones) digit is 3. The hundreds and tens digits, in that order, form a two-digit
integer that is divisible by 5. What is the five-digit integer? 2018 2a
8. The positive integers 34 and 80 have exactly two positive common divisors, namely 1 and 2. How
many positive integers n with 1 ≤ n ≤ 30 have the property that n and 80 have exactly two positive
common divisors? 2018 4a
9.
2018 6a
10. How many positive integers n satisfy 5 < 2 n < 2017? 2017 1b
11.
2017 2b&c
Currency:
1. Jimmy bought 600 Euros at the rate of 1 Euro equals $1.50. He then converted his 600 Euros back into
dollars at the rate of $1.00 equals 0.75 Euros. How many fewer dollars did Jimmy have after these two
transactions than he had before these two transactions? 2017 1c
Area & Perimeter:
1. In the diagram, two small circles of radius 1 are tangent to each other and to a larger circle of radius 2.
What is the area of the shaded region? 2019 2a
2.
2018 2c
3. The perimeter of equilateral 4P QR is 12. The perimeter of regular hexagon ST UV W X is also 12.
What is the ratio of the area of 4P QR to the area of ST UV W X? 2018 5a
4.
2017 3a
5 (Perimeter):
2017 3b
6. One of the faces of a rectangular prism has area 27 cm2 . Another face has area 32 cm2 . If the
volume of the prism is 144 cm3 , determine the surface area of the prism in cm2 . 2017 3c
Speed:
1. Kari jogs at a constant speed of 8 km/h. Mo jogs at a constant speed of 6 km/h. Kari and Mo jog from
the same starting point to the same finishing point along a straight road. Mo starts at 10:00 a.m. Kari
and Mo both finish at 11:00 a.m. At what time did Kari start to jog? 2019 2b
2. Linh is driving at 60 km/h on a long straight highway parallel to a train track. Every 10 minutes, she is
passed by a train travelling in the same direction as she is. These trains depart from the station behind
her every 3 minutes and all travel at the same constant speed. What is the constant speed of the trains,
in km/h? 2017 7a
What is the value of:
1. Michelle calculates the average of the following numbers: 5, 10, 15, 16, 24, 28, 33, 37 Daphne
removes one number and calculates the average of the remaining numbers. The average that Daphne
calculates is one less than the average that Michelle calculates. Which number does Daphne remove?
2019 3a
2. If 16 15 x = 32 4 3 , what is the value of x? 2019 3b
3. Suppose that 2 2022 + 2a 2 2019 = 72. Determine the value of a. 2019 3c
4.
5.
2019 6a
2018 3a
6. The line with equation x+ 3y = 7 is parallel to the line with equation y = mx+b. The line with equation y
= mx + b passes through the point (9, 2). Determine the value of b. 2019 2c
7.
2017 2a
8. The equations y = a(x − 2)(x + 4) and y = 2(x − h) 2 + k represent the same parabola. What are the
values of a, h and k? 2017 4a
Ratios:
1. Eleanor has 100 marbles, each of which is black or gold. The ratio of the number of black marbles to
the number of gold marbles is 1 : 4. How many gold marbles should she add to change this ratio to 1 : 6?
2020 2(b)
Finding the length:
1. Donna has a laser at C. She points the laser beam at the point E. The beam reflects off of DF at E and
then off of F H at G, as shown, arriving at point B on AD. If DE = EF = 1 m, what is the length of BD, in
metres? 2020 3(a)
2. Rectangle ABCD has AB = 4 and BC = 6. The semi-circles with diameters AE and F C each have radius r,
have centres S and T, and touch at a single point P, as shown. What is the value of r? 2020 6(a)
3. In the diagram, 4ABE is right-angled at A, 4BCD is right-angled at C, ∠ABC = 135◦ , and AB = AE = 7√ 2.
If DC = 4x, DB = 8x and DE = 8x − 6 for some real number x, determine all possible values of x. 2020 6(b)
4.
5.
6.
2019 4a
2019 8a
2018 2b
6. Three microphones A, B and C are placed on a line such that A is 1 km west of B and C is 2 km east of
B. A large explosion occurs at a point P not on this line. Each of the three microphones receives the
sound. The sound travels at 1 3 km/s. Microphone B receives the sound first, microphone A receives the
sound 1 2 s later, and microphone C receives it 1 s after microphone A. Determine the distance from
microphone B to the explosion at P. 2020 8b
7.
8.
2018 5b
2018 8b
Circle:
1. The points A(d, −d) and B(−d + 12, 2d − 6) both lie on a circle centered at the origin. Determine the
possible values of d. 2019 4b
2.
2017 8a
Volumes:
2017 8b
Permutation:
2017 9a
Sequences:
1. A geometric sequence has first term 10 and common ratio 1 2 . An arithmetic sequence has first term
10 and common difference d. The ratio of the 6th term in the geometric sequence to the 4th term in the
geometric sequence equals the ratio of the 6th term in the arithmetic sequence to the 4th term in the
arithmetic sequence. Determine all possible values of d. 2020 4(b)
2. Kerry has a list of n integers a1, a2, . . . , an satisfying a1 ≤ a2 ≤ . . . ≤ an. Kerry calculates the pairwise
sums of all m = 1 2 n(n − 1) possible pairs of integers in her list and orders these pairwise sums as s1 ≤ s2
≤ . . . ≤ sm. For example, if Kerry’s list consists of the three integers 1, 2, 4, the three pairwise sums are 3,
5, 6. (a) Suppose that n = 4 and that the 6 pairwise sums are s1 = 8, s2 = 104, s3 = 106, s4 = 110, s5 =
112, and s6 = 208. Determine two possible lists a1, a2, a3, a4 that Kerry could have. (b) Suppose that n =
5 and that the 10 pairwise sums are s1, s2, . . . , s10. Prove that there is only one possibility for Kerry’s
list a1, a2, a3, a4, a5. (c) Suppose that n = 16. Prove that there are two different lists a1, a2, . . . , a16 and
b1, b2, . . . , b16 that produce the same list of sums s1, s2, . . . , s120. 2020 10.
3.
2019 6b
4. In an arithmetic sequence with 5 terms, the sum of the squares of the first 3 terms equals the sum of
the squares of the last 2 terms. If the first term is 5, determine all possible values of the fifth term.2017
4b
Perfect Square:
1. Dan was born in a year between 1300 and 1400. Steve was born in a year between 1400 and 1500.
Each was born on April 6 in a year that is a perfect square. Each lived for 110 years. In what year while
they were both alive were their ages both perfect squares on April 7? 2017 5a
Right Angle Triangle:
1. Determine all values of k for which the points A(1, 2), B(11, 2) and C(k, 6) form a right-angled triangle.
2017 5b
Shortest Length Possible:
2017 6a
Trig:
1. If cos θ = tan θ, determine all possible values of sin θ, giving your answer(s) as simplified exact
numbers. 2017 6b
Functions:
1. For each positive real number x, define f(x) to be the number of prime numbers p that satisfy x ≤ p ≤ x
+ 10. What is the value of f(f(20))? 2020 5(a).
2. Suppose that the function g satisfies g(x) = 2x − 4 for all real numbers x and that g −1 is the inverse
function of g. Suppose that the function f satisfies g(f(g −1 (x))) = 2x 2 + 16x + 26 for all real numbers x.
What is the value of f(π)? 2020 7(a)
3. Consider the function f(x) = x 2 − 2x. Determine all real numbers x that satisfy the equation f(f(f(x))) =
3. 2019 7b
4. For positive integers a and b, define f(a, b) = a b + b a + 1 ab . For example, the value of f(1, 2) is 3. (a)
Determine the value of f(2, 5). (b) Determine all positive integers a for which f(a, a) is an integer. (c) If a
and b are positive integers and f(a, b) is an integer, prove that f(a, b) must be a multiple of 3. (d)
Determine four pairs of positive integers (a, b), with 2 < a < b, for which f(a, b) is an integer. 2019 9
5. A function f is defined so that • f(1) = 1, • if n is an even positive integer, then f(n) = f 1 2 n , and • if n
is an odd positive integer with n > 1, then f(n) = f(n − 1) + 1. For example, f(34) = f(17) and f(17) = f(16) +
1. Determine the value of f(50) 2018 4b
System of Equations (Log, pairs, satisy):
1. Determine all triples (x, y, z) of real numbers that satisfy the following system of equations: (x − 1)(y −
2) = 0 (x − 3)(z + 2) = 0 x + yz = 9 2020 5(b).
2. Determine all pairs of angles (x, y) with 0◦ ≤ x < 180◦ and 0◦ ≤ y < 180◦ that satisfy the following system
of equations: log2 (sin x cos y) = − 3 2 log2 sin x cos y = 1 2 2020 7(b)
3. Consider the system of equations: c + d = 2000 c d = k Determine the number of integers k with k ≥ 0
for which there is at least one pair of integers (c, d) that is a solution to the system. 2019 5b
4. Determine all real numbers x for which 2 log2 (x − 1) = 1 − log2 (x + 2). 2019 7a
5. Suppose that x satisfies 0 < x < π 2 and cos 3 2 cos x = sin 3 2 sin x . Determine all possible values of
sin 2x, expressing your answers in the form aπ2 + bπ + c d where a, b, c, d are integers. 2019 8b.
2018 3
6. Determine all values of x such that log2x (48√3 3) = log3x (162√3 2). 2018 8a
7.
2017 7b
Probability:
1. Four tennis players Alain, Bianca, Chen, and Dave take part in a tournament in which a total of three
matches are played. First, two players are chosen randomly to play each other. The other two players
also play each other. The winners of the two matches then play to decide the tournament champion.
Alain, Bianca and Chen are equally matched (that is, when a match is played between any two of them,
the probability that each player wins is 1 2 ). When Dave plays each of Alain, Bianca and Chen, the
probability that Dave wins is p, for some real number p. Determine the probability that Bianca wins the
tournament, expressing your answer in the form ap2 + bp + c d where a, b, c, and d are integers 2020 8a
2.
2019 10
3. Eight people, including triplets Barry, Carrie and Mary, are going for a trip in four canoes. Each canoe
seats two people. The eight people are to be randomly assigned to the four canoes in pairs. What is the
probability that no two of Barry, Carrie and Mary will be in the same canoe? 2018 7a
Diagonal/Square
1. Diagonal W Y of square W XY Z has slope 2. Determine the sum of the slopes of W X and XY . 2018 7b
Misc:
2020 3(b)
2020 9
2018 6b
2018 9
2018 10
2017 2b
2017 10