Sample Individual Questions 2012 Canadian Team Mathematics Contest (CTMC)

The CENTRE for EDUCATION in MATHEMATICS and COMPUTING
presents the
2012 Canadian Team Mathematics Contest (CTMC)
Sample Individual Questions
In this event, participants work individually and submit one answer per question.
1. If x is odd, which of the following is an integer?
(A)
x
2
(B)
x−2
2
(C)
x3
23
(D)
(x + 1)2
2
(E)
3x2 − 2
2
2. The lengths of two of the three sides of a right-angled triangle are 15 and 36. Determine the
possible lengths of the third side.
3. Consider a 4 by 8 rectangle ABCD.
Let P be the midpoint of CD. Determine the exact distance from P to one of the diagonals.
4. What is the smallest positive integer n divisible by each of 2, 4, 6, 8, 10, 12, 14, 16, 18, and
20?
5. How many points (x, y), with x and y both integers, are on or inside the circle x2 + y 2 = 100?
6. For integers a and b, we define a ∗ b = ab + ba . If 2 ∗ x = 100, determine the value of x.
7. A sequence is called geometric with ratio r if each term after the first is equal to r times the
term before it. For example, the sequence {3, 15, 75, 375, . . . } is a geometric sequence with
ratio 5.
A different geometric sequence has a first term c and ratio r where c and r are both positive
integers. The average of the fourth and eighth terms in the sequence is 2720. Determine the
value c.
8. The symbol bxc denotes the greatest integer less than or equal to x. For example, b3c = 3,
b5.7c = 5. If n is an integer less than 64 and
√
√
√
√
3
3
3
b 1c + b 2c + b 3c + ... + b 3 nc = 2n,
determine the value of n.
9. A very large square room is tiled with black squares and white squares,
with each tile measuring 1 metre by 1 metre. The tiles are laid using
the following pattern, that continues down and to the right through the
whole room. If the ratio of one colour’s area to the other colour’s area
is 247:260, determine the dimensions of the room.
10. Consider the following construction: numbers a1 , a2 , a3 , . . . , ak are chosen and k corresponding
squares are drawn so that:
• 0 < a1 < a2 < a3 < ... < ak .
• The ith square has its bottom left vertex at the origin and its top right vertex at (ai , ai ).
• The squares define mutually exclusive regions with areas A1 , A2 , A3 , . . . , Ak , and
A1 = A2 = A3 = · · · = Ak .
The diagram shows such a construction with k = 5, but the construction could be extended to
much higher values of k.
If a9 and a36 are consecutive integers, determine the smallest t > 2010 such that at is an
integer.
The CENTRE for EDUCATION in MATHEMATICS and COMPUTING
presents the
2012 Canadian Team Mathematics Contest (CTMC)
Sample Relay Questions
In this event, the team is split into two groups of three.
Group member A completes part (a) and passes the answer to the right. Group member B needs that
answer (TNYWR, “the number you will receive”) to complete part (b). Similarly, group member C
needs an answer from B to complete part (c).
Relay #1 (a) The ratio of 3x − 4 to y + 15 is constant. When y = 3, x will equal 2. What will x equal
when y = 12?
(b) Let t = T N Y W R.
If k = 30t, determine the sum of the positive integers from 1 to k inclusive.
(c) Let t = T N Y W R.
2F (n) + 1
for n = 1, 2, 3, . . . and F (1) = 2. Determine the value of F (t).
F (n + 1) =
2
Relay #2 (a) If a ∗ b =
ab
, determine the value of (4 ∗ 4) ∗ 4.
a+b
(b) Let t = T N Y W R. Let k = 3t.
The convex area bounded by x = 1, x = k, the x-axis and the line y = mx + 4 is 17.
Determine the value of m.
(c) Let t = T N Y W R. Let k = 15t.
Let n represent the number of ways that k dollars can be changed into dimes and quarters,
with at least one of each coin being used. Determine the value of n.
Relay #3 (a) Let x and y be positive integers such that 3x + 5y = 100. If S = x + y, what is the largest
possible value of S?
(b) Let t = T N Y W R.
Determine the sum of the interior angles of a polygon that has t sides.
(c) Let t = T N Y W R.
An arithmetic sequence has 170 as the first term and 370 as the last term. The sum of
the terms in the sequence is t. How many terms are in the sequence?
The CENTRE for EDUCATION in MATHEMATICS and COMPUTING
presents the
2012 Canadian Team Mathematics Contest (CTMC)
Sample Team Questions
In this event, the team collaborates to submit at most one answer per question.
1. Find all positive integers x less than 15 such that 5x = 3n + 4 for some integer n.
2. There are five numbers v, w, x, y, z, with v < w < x < y < z. When taken four at a time, their
sums are 37, 42, 48, 51, and 58. Determine v, w, x, y, z.
1
(1 + 2 + 3 + · · · + 16).
3. Find the sum of 1 + 12 (1 + 2) + 31 (1 + 2 + 3) + 14 (1 + 2 + 3 + 4) + · · · + 16
4. A gumball machine contains gumballs that are blue, red or yellow. There are half as many blue
gumballs as red gumballs. There are three times as many red gumballs as yellow gumballs. If
the total number of gumballs is 99, determine the number of red gumballs.
5. Given a right-angled triangle P QR where angle Q is 90◦ and with sides QR = 5 and P Q = 12.
A square is constructed on each side P Q, QR, and P R. Label these squares AP QF, QRDE
and P BCR. A hexagon is formed by joining AB, CD, and EF . Find the area of hexagon
ABCDEF .
6. Find the sum of all values of θ, 0 ≤ θ ≤ 3π, such that sin(θ) = cos(2θ).
7. Let r, s, and t be the roots of 0 = x3 + 2x2 + x − 1, with r, s, t ∈ R. Evaluate r2 + s2 + t2 , with
r, s, t ∈ R
8. How many numbers are there that are divisible by 9 and have digits of the form ABCABCABC
with A 6= B 6= C and A 6= 0?
9. Suppose that 4ABC has vertices A(3, 0), B(0, 3) and C, where C is on the line x + y = 7.
Find the area of 4ABC.
10. An equilateral triangle has side length 5. You are given an unlimited number of circles of radius
1 that can be placed inside the triangle. The circles may overlap each other, but they must
remain entirely inside the triangle. What is the maximum area that can be covered by the
circles?
11. A gardener plants three oak trees, four birch trees, and five maple trees in a row. He plants
them in a random order, each arrangement being equally likely. Find the probability that no
two maple trees are next to one another.
12. Find the solutions (x, y) to the following system of equations:
|x| + |y| = 4
xy − 3x + y = 4
13. The number n = 1764622B5362A6 is divisible by 792. A and B represent digits. Determine
the sum A + B.
14. Solve for (x, y, z) where
xy
= 17,
x+y
yz
1
= ,
y+z
8
xz
1
= .
x+z
7
15. Find x + y if x, y are positive integers such that xy + x + y = 71 and x2 y + xy 2 = 880.
16. In the diagram, the parabola y = x2 has an inscribed isosceles 4AOB where AB is parallel to
the x-axis. If AO = AB, determine the length of AO.
17. Two equations
x3 + 5x2 + px + q = 0
x3 + 7x2 + px + r = 0
have two roots in common. If the third roots of each equation are a and b respectively, determine
the ordered pair (a, b).
18. In how many different ways can 48 be expressed as the sum of two prime numbers p and q
with p < q?
19. If log2 x2 + log 1 x = 5, compute x.
2
20. For how many integers n is 0 < n3 − n2 − n < 890?
21. Solve for (x, y) if
2x2 + 2xy + y 2 = 73
x2 + xy + 2y 2 = 74
22. Sn and Tn are the respective sums of two arithmetic series.
Sn
7n + 1
=
Tn
4n + 27
for all n. Determine the ratio of the eleventh term of the first series to the eleventh term of the
second series.
23. Find n such that n is a positive integer and 2(22 ) + 3(23 ) + 4(24 ) + · · · + n(2n ) = 2n+10 .
24. Find the solutions to z 2 = 7 − 24i, where z ∈ C.
25. A partition of a positive integer N is a non-decreasing sequence of positive integers which sum
to N - that is, N = n1 + n2 + n3 + · · · + ns where n1 ≤ n2 ≤ n3 ≤ · · · ≤ ns .
For example the partitions of 4 are:
4=1+1+1+1
4=1+1+2
4=1+3
4=2+2
4=4
An odd partition is a partition in which each term ni is odd. Find the number of odd partitions
of 21.