AoPS Community 2022 Canadian Mathematical Olympiad Qualification

Canadian Mathematical Olympiad Qualification Repechage 2022

www.artofproblemsolving.com/community/c3173605
by Dankster42, parmenides51, vanlucas

1 Let n ≥ 2 be a positive integer. On a spaceship, there are n crewmates. At most one accusation
of
being an imposter can occur from one crewmate to another crewmate. Multiple accusations
are
thrown, with the following properties:
• Each crewmate made a different number of accusations.
• Each crewmate received a different number of accusations.
• A crewmate does not accuse themself.
Prove that no two crewmates made accusations at each other.

2 Determine all pairs of integers (m, n) such that m2 + n and n2 + m are both perfect squares.

3 Consider n real numbers x0 , x1 , ..., xn−1 for an integer n ≥ 2. Moreover, suppose that for any
integer i, xi+n = xi . Prove that
n−1
X
xi (3xi − 4xi+1 + xi+2 ) ≥ 0.
i=0

4 For a non-negative integer n, call a one-variable polynomial F with integer coefficients n-good
if:
(a) F (0) = 1
(b) For every positive integer c, F (c) > 0, and
(c) There exist exactly n values of c such that F (c) is prime.
Show that there exist infinitely many non-constant polynomials that are not n-good for any n.

5 Alice has four boxes, 327 blue balls, and 2022 red balls. The blue balls are labeled 1 to 327. Alice
first puts each of the balls into a box, possibly leaving some boxes empty. Then, a random label
between 1 and 327 (inclusive) is selected, Alice finds the box the ball with the label is in, and
selects a random ball from that box. What is the maximum probability that she selects a red
ball?

6 Let a, b, c be real numbers, which are not all equal, such that
1 1 1
a+b+c= + + = 3.
a b c
Prove that at least one of a, b, c is negative.

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7 Let ABC be a triangle with |AB| < |AC|, where || denotes length. Suppose D, E, F are points
on side BC such that D is the foot of the perpendicular on BC from A, AE is the angle bisector
of ∠BAC, and F is the midpoint of BC. Further suppose that ∠BAD = ∠DAE = ∠EAF =
∠F AC. Determine all possible values of ∠ABC.

8 Let {m, n, k} be positive integers. {k} coins are placed in the squares of an m × n grid. A square
may contain any number of coins, including zero. Label the {k} coins C1 , C2 , Ck . Let ri be the
number of coins in the same row as Ci , including Ci itself. Let si be the number of coins in the
same column as Ci , including Ci itself. Prove that
k
X 1 m+n
≤
ri + si 4
i=1

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AoPS Community 2021 Canadian Mathematical Olympiad Qualification

Canadian Mathematical Olympiad Qualification Repechage 2021

www.artofproblemsolving.com/community/c1976896
by Audiophile

1 Determine all real polynomials p such that p(x + p(x)) = x2 p(x) for all x.

2 Determine all integer solutions to the system of equations:

xy + yz + zx = −4
x2 + y 2 + z 2 = 24
x3 + y 3 + z 3 + 3xyz = 16

3 ABCDE is a regular pentagon. Two circles C1 and C2 are drawn through B with centers A and
C respectively. Let the other intersection of C1 and C2 be P . The circle with center P which
passes through E and D intersects C2 at X and AE at Y . Prove that AX = AY .

4 Let O be the centre of the circumcircle of triangle ABC and let I be the centre of the incircle
of triangle ABC. A line passing through the point I is perpendicular to the line IO and passes
through the incircle at points P and Q. Prove that the diameter of the circumcircle is equal to
the perimeter of triangle OP Q.

5 Alphonse and Beryl are playing a game. The game starts with two rectangles with integer side
lengths. The players alternate turns, with Alphonse going first. On their turn, a player chooses
one rectangle, and makes a cut parallel to a side, cutting the rectangle into two pieces, each
of which has integer side lengths. The player then discards one of the three rectangles (either
the one they did not cut, or one of the two pieces they cut) leaving two rectangles for the other
player. A player loses if they cannot cut a rectangle.
Determine who wins each of the following games:
(a) The starting rectangles are 1 × 2020 and 2 × 4040.
(b) The starting rectangles are 100 × 100 and 100 × 500.

6 Show that (w, x, y, z) = (0, 0, 0, 0) is the only integer solution to the equation

w2 + 11x2 − 8y 2 − 12yz − 10z 2 = 0

7 If A, B and C are real angles such that

cos(B − C) + cos(C − A) + cos(A − B) = −3/2,

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find
cos(A) + cos(B) + cos(C)

8 King Radford of Peiza is hosting a banquet in his palace. The King has an enormous circular
table with 2021 chairs around it. At The King’s birthday celebration, he is sitting in his throne
(one of the 2021 chairs) and the other 2020 chairs are filled with guests, with the shortest guest
sitting to the King’s left and the remaining guests seated in increasing order of height from
there around the table. The King announces that everybody else must get up from their chairs,
run around the table, and sit back down in some chair. After doing this, The King notices that
the person seated to his left is different from the person who was previously seated to his left.
Each other person at the table also notices that the person sitting to their left is different.
Find a closed form expression for the number of ways the people could be sitting around the
table at the end. You may use the notation Dn , the number of derangements of a set of size
n, as part of your expression.

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AoPS Community 2020 Canadian Mathematical Olympiad Qualification

Canadian Mathematical Olympiad Qualification Repechage 2020

www.artofproblemsolving.com/community/c1135962
by parmenides51

√ √ √ √
1 Show that for all integers a ≥ 1,b a + a + 1 + a + 2c = b 9a + 8c

2 Given a set S, of integers, an optimal partition of S into sets T, U is a partition which minimizes
the value |t − u|, where t and u are the sum of the elements of T and U respectively.
Let P be a set of distinct positive integers such that the sum of the elements of P is 2k for a
positive integer k, and no subset of P sums to k.
Either show that there exists such a P with at least 2020 different optimal partitions, or show
that such a P does not exist.

3 Let N be a positive integer and A = a1 , a2 , ..., aN be a sequence of real numbers.

Define the sequence f (A) to be
 
a1 + a2 a2 + a3 aN −1 + aN aN + a1
f (A) = , , ..., ,
2 2 2 2

and for k a positive integer define f k (A) to bef applied to A consecutively k times (i.e. f (f (...f (A))))
Find all sequences A = (a1 , a2 , ..., aN ) of integers such that f k (A) contains only integers for
all k.

4 Determine all graphs G with the following two properties: • G contains at least one Hamilton
path. • For any pair of vertices, u, v ∈ G, if there is a Hamilton path from u to v then the edge
uv is in the graph G

5 We define the following sequences:

Sequence A has an = n.
Sequence B has bn = P an when an 6≡ 0 (mod 3) and bn = 0 otherwise.
Sequence C has cn = ni=1 bi
. Sequence D has dn =Pcn when cn 6≡ 0 (mod 3) and dn = 0 otherwise.
Sequence E has en = ni=1 di
Prove that the terms of sequence E are exactly the perfect cubes.

6 In convex pentagon ABCDE, AC is parallel to DE, AB is perpendicular to AE, and BC is

perpendicular to CD. If H is the orthocentre of triangle ABC and M is the midpoint of segment
DE, prove that AD, CE and HM are concurrent.

7 Let a, b, c be positive real numbers with ab + bc + ac = abc. Prove that

bc ac ab 1
+ + c+1 ≥
aa+1 bb+1 c 3

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√
8 Find all pairs (a, b) of positive rational numbers such that b
a = ab

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AoPS Community 2018 Canadian Mathematical Olympiad Qualification

Canadian Mathematical Olympiad Qualification Repechage 2018

www.artofproblemsolving.com/community/c639354
by cjquines0

1 Determine all real solutions to the following system of equations:

(
y = 4x3 + 12x2 + 12x + 3
x = 4y 3 + 12y 2 + 12y + 3.

2 We call a pair of polygons, p and q, nesting if we can draw one inside the other, possibly after
rotation and/or reflection; otherwise we call them non-nesting.
Let p and q be polygons. Prove that if we can find a polygon r, which is similar to q, such that
r and p are non-nesting if and only if p and q are not similar.

3 Let ABC be a triangle with AB = BC. Prove that 4ABC is an obtuse triangle if and only if
the equation
Ax2 + Bx + C = 0
has two distinct real roots, where A, B, C, are the angles in radians.

4 Construct a convex polygon such that each of its sides has the same length as one of its
diagonals and each diagonal has the same length as one of its sides, or prove that such a
polygon does not exist.

5 A palindrome is a number that remains the same when its digits are reversed. Let n be a prod-
uct of distinct primes not divisible by 10. Prove that infinitely many multiples of n are palin-
dromes.

6 Let n ≥ 2 be a positive integer.PDetermineQthe number of n-tuples (x1 , x2 , . . . , xn ) such that

xk ∈ {0, 1, 2} for 1 ≤ k ≤ n and nk=1 xk − nk=1 xk is divisible by 3.

7 Let n be a positive integer, with prime factorization

n = pe11 pe22 · · · perr
for distinct primes p1 , . . . , pr and ei positive integers. Define
rad(n) = p1 p2 · · · pr ,
the product of all distinct prime factors of n.
Find all polynomials P (x) with rational coefficients such that there exists infinitely many pos-
itive integers n with P (n) = rad(n).

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8 Let n and k be positive integers with 1 ≤ k ≤ n. A set of cards numbered 1 to n are arranged
randomly in a row from left to right. A person alternates between performing the following
moves:

- The leftmost card in the row is moved k − 1 positions to the right while the cards in positions
2 through k are each moved one place to the left.
- The rightmost card in the row is moved k − 1 positions to the left while the cards in positions
n − k + 1 through n − 1 are each moved one place to the right.
Determine the probability that after some number of moves the cards end up in order from 1
to n, left to right.

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AoPS Community 2017 Canadian Mathematical Olympiad Qualification

Canadian Mathematical Olympiad Qualification Repechage 2017

www.artofproblemsolving.com/community/c438033
by cjquines0

1 Malcolm writes a positive integer on a piece of paper. Malcolm doubles this integer and sub-
tracts 1, writing this second result on the same piece of paper. Malcolm then doubles the sec-
ond integer and adds 1, writing this third integer on the paper. If all of the numbers Malcolm
writes down are prime, determine all possible values for the first integer.

2 For any positive integer n, let ϕ(n) be the number of integers in the set {1, 2, . . . , n} whose
greatest common divisor with n is 1. Determine the maximum value of ϕ(n) n
for n in the set
{2, . . . , 1000} and all values of n for which this maximum is attained.

3 Determine all functions f : R → R that satisfy the following equation for all x, y ∈ R.
(x + y)f (x − y) = f (x2 − y 2 ).

4 In this question we re-define the operations addition and multiplication as follows: a + b is

defined as the minimum of a and b, while a ∗ b is defined to be the sum of a and b. For example,
3 + 4 = 3, 3 ∗ 4 = 7, and
3 ∗ 42 + 5 ∗ 4 + 7 = min(3 plus 4 plus 4, 5 plus 4, 7) = min(11, 9, 7) = 7.
Let a, b, c be real numbers. Characterize, in terms of a, b, c, what the graph of y = ax2 + bx + c
looks like.

5 Prove for all real numbers x, y,

(x2 + 1)(y 2 + 1) + 4(x − 1)(y − 1) ≥ 0.
Determine when equality holds.

6 Let N be a positive integer. There are N tasks, numbered 1, 2, 3, . . . , N , to be completed. Each

task takes one minute to complete and the tasks must be completed subjected to the following
conditions:
- Any number of tasks can be performed at the same time.
- For any positive integer k, task k begins immediately after all tasks whose numbers are divi-
sors of k, not including k itself, are completed.
- Task 1 is the first task to begin, and it begins by itself.
Suppose N = 2017. How many minutes does it take for all of the tasks to complete? Which
tasks are the last ones to complete?

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7 Given a set Sn = {1, 2, 3, . . . , n}, we define a preference list to be an ordered subset of Sn . Let
Pn be the number of preference lists of Sn . Show that for positive integers n > m, Pn − Pm is
divisible by n − m.
[i]Note: the empty set and Sn are subsets of Sn .[/i]

8 A convex quadrilateral ABCD is said to be dividable if for every internal point P , the area
of 4P AB plus the area of 4P CD is equal to the area of 4P BC plus the area of 4P DA.
Characterize all quadrilaterals which are dividable.

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AoPS Community 2016 Canadian Mathematical Olympiad Qualification

Canadian Mathematical Olympiad Qualification Repechage 2016

www.artofproblemsolving.com/community/c286003
by cjquines0

1 (a) Find all positive integers n such that 11|(3n + 4n ).

(b) Find all positive integers n such that 31|(4n + 7n + 20n ).

2 Let P = (7, 1) and let O = (0, 0).

(a) If S is a point on the line y = x and T is a point on the horizontal x-axis so that P is on the
line segment ST , determine the minimum possible area of triangle OST .
(b) If U is a point on the line y = x and V is a point on the horizontal x-axis so that P is on the
line segment U V , determine the minimum possible perimeter of triangle OU V .

3 Given an n × n × n grid of unit cubes, a cube is good if it is a sub-cube of the grid and has
side length at least two. If a good cube contains another good cube and their faces do not
intersect, the first good cube is said to properly contain the second. What is the size of the
largest possible set of good cubes such that no cube in the set properly contains another
cube in the set?

4 Determine all functions f : R → R such that

f (x + f (y)) + f (x − f (y)) = x.

5 Consider a convex polygon P with n sides and perimeter P0 . Let the polygon Q, whose vertices
are the midpoints of the sides of P , have perimeter P1 . Prove that P1 ≥ P20 .

6 Determine all ordered triples of positive integers (x, y, z) such that gcd(x + y, y + z, z + x) >
gcd(x, y, z).

7 Starting at (0, 0), Richard takes 2n + 1 steps, with each step being one unit either East, North,
West, or South. For each step, the direction is chosen uniformly at random from the four pos-
sibilities. Determine the probability that Richard ends at (1, 0).

8 Let n ≥ 3 be a positive integer. A [i]chipped n-board[/i] is a 2 × n checkerboard with the bottom

left square removed. Lino wants to tile a chipped n-board and is allowed to use the following
types of tiles:
- Type 1: any 1 × k board where 1 ≤ k ≤ n

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- Type 2: any chipped k-board where 1 ≤ k ≤ n that must cover the left-most tile of the 2 × n
checkerboard.
Two tilings T1 and T2 are considered the same if there is a set of consecutive Type 1 tiles in
both rows of T1 that can be vertically swapped to obtain the tiling T2 . For example, the following
three tilings of a chipped 7-board are the same:
http://i.imgur.com/8QaSgc0.png
For any positive integer n and any positive integer 1 ≤ m ≤ 2n − 1, let cm,n be the number of
distinct tilings of a chipped n-board using exactly m tiles (any combination of tile types may
be used), and define the polynomial
2n−1
X
Pn (x) = cm,n xm .
m=1

Find, with justification, polynomials f (x) and g(x) such that

Pn (x) = f (x)Pn−1 (x) + g(x)Pn−2 (x)

for all n ≥ 3.

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AoPS Community 2015 Canadian Mathematical Olympiad Qualification

Canadian Mathematical Olympiad Qualification Repechage 2015

www.artofproblemsolving.com/community/c285832
by cjquines0

1 Find all integer solutions to the equation 7x2 y 2 + 4x2 = 77y 2 + 1260.

2 A polynomial f (x) with integer coefficients is said to be tri-divisible if 3 divides f (k) for any
integer k. Determine necessary and sufficient conditions for a polynomial to be tri-divisible.

3 Let N be a 3-digit number with three distinct non-zero digits. We say that N is mediocre if it
has the property that when all six 3-digit permutations of N are written down, the average is
N . For example, N = 481 is mediocre, since it is the average of {418, 481, 148, 184, 814, 841}.
Determine the largest mediocre number.

4 Given an acute-angled triangle ABC whose altitudes from B and C intersect at H, let P be
any point on side BC and X, Y be points on AB, AC, respectively, such that P B = P X and
P C = P Y . Prove that the points A, H, X, Y lie on a common circle.

5 Let x and y be positive real numbers such that x + y = 1. Show that

 2  2
x+1 y+1
+ ≥ 18.
x y

6 Let 4ABC be a right-angled triangle with ∠A = 90◦ , and AB < AC. Let points D, E, F be
located on side BC such that AD is the altitude, AE is the internal angle bisector, and AF is
the median.
Prove that 3AD + AF > 4AE.

7 A (0x , 1y , 2z )-string is an infinite ternary string such that:

- If there is a 0 in position i then there is a 1 in position i + x,
- if there is a 1 in position j then there is a 2 in position j + y,
- if there is a 2 in position k then there is a 0 in position k + z.
For how many ordered triples of positive integers (x, y, z) with x, y, z ≤ 100 does there exist
(0x , 1y , 2z )-string?

8 A magical castle has n identical rooms, each of which contains k doors arranged in a line. In
room i, 1 ≤ i ≤ n − 1 there is one door that will take you to room i + 1, and in room n there is

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one door that takes you out of the castle. All other doors take you back to room 1. When you
go through a door and enter a room, you are unable to tell what room you are entering and you
are unable to see which doors you have gone through before. You begin by standing in room 1
and know the values of n and k. Determine for which values of n and k there exists a strategy
that is guaranteed to get you out of the castle and explain the strategy. For such values of n
and k, exhibit such a strategy and prove that it will work.

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AoPS Community 2014 Canadian Mathematical Olympiad Qualification

Canadian Mathematical Olympiad Qualifaction Repechage 2014

www.artofproblemsolving.com/community/c285739
by cjquines0

1 Let f : Z → Z+ be a function, and define h : Z × Z → Z+ by h(x, y) = gcd(f (x), f (y)). If h(x, y)

is a two-variable polynomial in x and y, prove that it must be constant.

2 Alphonse and Beryl play a game involving n safes. Each safe can be opened by a unique key
and each key opens a unique safe. Beryl randomly shuffles the n keys, and after placing one
key inside each safe, she locks all of the safes with her master key. Alphonse then selects m of
the safes (where m < n), and Beryl uses her master key to open just the safes that Alphonse
selected. Alphonse collects all of the keys inside these m safes and tries to use these keys to
open up the other n−m safes. If he can open a safe with one of the m keys, he can then use the
key in that safe to try to open any of the remaining safes, repeating the process until Alphonse
successfully opens all of the safes, or cannot open any more. Let Pm (n) be the probability that
Alphonse can eventually open all n safes starting from his initial selection of m keys.
(a) Show that P2 (3) = 23 .
(b) Prove that P1 (n) = n1 .
(c) For all integers n ≥ 2, prove that
2 n−2
P2 (n) = · P1 (n − 1) + · P2 (n − 1).
n n

(d) Determine a formula for P2 (n).

3 Let 1000 ≤ n = ABCD10 ≤ 9999 be a positive integer whose digits ABCD satisfy the divisibility
condition:
1111|(ABCD + AB × CD).
Determine the smallest possible value of n.

4 In 4ABC, the interior sides of which are mirrors, a laser is placed at point A1 on side BC. A
laser beam exits the point A1 , hits side AC at point B1 , and then reflects off the side. (Because
this is a laser beam, every time it hits a side, the angle of incidence is equal to the angle of
reflection). It then hits side AB at point C1 , then side BC at point A2 , then side AC again at
point B2 , then side AB again at point C2 , then side BC again at point A3 , and finally, side AC
again at point B3 .
(a) Prove that ∠B3 A3 C = ∠B1 A1 C.
(b) Prove that such a laser exists if and only if all the angles in 4ABC are less than 90◦ .

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5 Let f (x) = x4 + 2x3 − x − 1.

(a) Prove that f (x) cannot be written as the product of two non-constant polynomials with
integer coefficients.
(b) Find the exact values of the 4 roots of f (x).

6 Given a triangle A, B, C, X is on side AB, Y is on side AC, and P and Q are on side BC such
that AX = AY, BX = BP and CY = CQ. Let XP and Y Q intersect at T . Prove that AT
passes through the midpoint of P Q.

7 A bug is standing at each of the vertices of a regular hexagon ABCDEF . At the same time
each bug picks one of the vertices of the hexagon, which it is not currently in, and immediately
starts moving towards that vertex. Each bug travels in a straight line from the vertex it was in
originally to the vertex it picked. All bugs travel at the same speed and are of negligible size.
Once a bug arrives at a vertex it picked, it stays there. In how many ways can the bugs move
to the vertices so that no two bugs are ever in the same spot at the same time?

8 For any given non-negative integer m, let f (m) be the number of 1’s in the base 2 representation
of m. Let n be a positive integer. Prove that the integer
n −1
2X  
f (m) m
(−1) ·2
m=0

contains at least n! positive divisors.

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AoPS Community 2013 Canadian Mathematical Olympiad Qualification
Repechage
Canadian Mathematical Olympiad Qualification Repechage 2013
www.artofproblemsolving.com/community/c4055
by TheMaskedMagician

1 Determine all real solutions to the following equation:

x x−1 +1)
2(2 ) − 3 · 2(2 + 8 = 0.

2 In triangle ABC, ∠A = 90◦ and ∠C = 70◦ . F is point on AB such that ∠ACF = 30◦ , and E is
a point on CA such that ∠CF E = 20◦ . Prove that BE bisects ∠B.

3 A positive integer n has the property that there are three positive integers x, y, z such that
lcm(x, y) = 180, lcm(x, z) = 900, and lcm(y, z) = n, where lcm denotes the lowest common
multiple. Determine the number of positive integers n with this property.

4 Four boys and four girls each bring one gift to a Christmas gift exchange. On a sheet of paper,
each boy randomly writes down the name of one girl, and each girl randomly writes down the
name of one boy. At the same time, each person passes their gift to the person whose name
is written on their sheet. Determine the probability that both of these events occur:
- (i) Each person receives exactly one gift;
- (ii) No two people exchanged presents with each other (i.e., if A gave his gift to B, then B did
not give her gift to A).

5 For each positive integer k, let S(k) be the sum of its digits. For example, S(21) = 3 and
S(105) = 6. Let n be the smallest integer for which S(n)−S(5n) = 2013. Determine the number
of digits in n.

6 Let x, y, z be real numbers that are greater than or equal to 0 and less than or equal to 1
2

- (a) Determine the minimum possible value of

x + y + z − xy − yz − zx

and determine all triples (x, y, z) for which this minimum is obtained.
- (b) Determine the maximum possible value of

x + y + z − xy − yz − zx

and determine all triples (x, y, z) for which this maximum is obtained.

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Repechage
7 Consider the following layouts of nine triangles with the letters A, B, C, D, E, F, G, H, I in its
interior.

C
B D
F H
E G I

A sequence of letters, each letter chosen fromA, B, C, D, E, F, G, H, I is said to be triangle-

friendly if the rst and last letter of the sequence is C, and for every letter except the rst letter,
the triangle containing this letter shares an edge with the triangle containing the previous
letter in the sequence. For example, the letter after C must be either A, B, or D. For example,
CBF BC is triangle-friendly, but CBF GH and CBBHC are not.
- (a) Determine the number of triangle-friendly sequences with 2012 letters.
- (b) Determine the number of triangle-friendly sequences with exactly 2013 letters.

8 Let 4ABC be an acute-angled triangle with orthocentre H and circumcentre O. Let R be the
radius of the circumcircle.

Let A0 be the point on AO (extended if necessary) for which HA0 ⊥ AO.

Let B 0 be the point on BO (extended if necessary) for which HB 0 ⊥ BO.
Let C 0 be the point on CO (extended if necessary) for which HC 0 ⊥ CO.

Prove that HA0 + HB 0 + HC 0 < 2R

(Note: The orthocentre of a triangle is the intersection of the three altitudes of the triangle.
The circumcircle of a triangle is the circle passing through the triangles three vertices. The
circummcentre is the centre of the circumcircle.)

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AoPS Community 2012 Canadian Mathematical Olympiad Qualification
Repechage
Canadian Mathematical Olympiad Qualification Repechage 2012
www.artofproblemsolving.com/community/c4054
by TheMaskedMagician

1 The front row of a movie theatre contains 45 seats.

- (a) If 42 people are sitting in the front row, prove that there are 10 consecutive seats that are
all occupied.
- (b) Show that this conclusion doesnt necessarily hold if only 41 people are sitting in the front
row.

2 Given a positive integer m, let d(m) be the number of positive divisors of m. Determine all
positive integers n such that d(n) + d(n + 1) = 5.

3 We say that (a, b, c) form a fantastic triplet if a, b, c are positive integers, a, b, c form a geometric
sequence, and a, b + 1, c form an arithmetic sequence. For example, (2, 4, 8) and (8, 12, 18) are
fantastic triplets. Prove that there exist infinitely many fantastic triplets.

4 Let ABC be a triangle such that ∠BAC = 90◦ and AB < AC. We divide the interior of the
triangle into the following six regions:

S1 = set of all points P inside 4ABC such that P A < P B < P C

S2 = set of all points P inside 4ABC such that P A < P C < P B
S3 = set of all points P inside 4ABC such that P B < P A < P C
S4 = set of all points P inside 4ABC such that P B < P C < P A
S5 = set of all points P inside 4ABC such that P C < P A < P B
S6 = set of all points P inside 4ABC such that P C < P B < P A

Suppose that the ratio of the area of the largest region to the area of the smallest non-empty
region is 49 : 1. Determine the ratio AC : AB.

5 Given a positive integer n, let d(n) be the largest positive divisor of n less than n. For example,
d(8) = 4 and d(13) = 1. A sequence of positive integers a1 , a2 , . . . satisfies

ai+1 = ai + d(ai ),

for all positive integers i. Prove that regardless of the choice of a1 , there are innitely many
terms in the sequence divisible by 32011 .

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6 Determine whether there exist two real numbers a and b such that both (x − a)3 + (x − b)2 + x
and (x − b)3 + (x − a)2 + x contain only real roots.

7 Six tennis players gather to play in a tournament where each pair of persons play one game,
with one person declared the winner and the other person the loser. A triplet of three players
{A, B , C } is said to be cyclic if A wins against B , B wins against C and C wins against A.

- (a) After the tournament, the six people are to be separated in two rooms such that none of
the two rooms contains a cyclic triplet. Prove that this is always possible.
- (b) Suppose there are instead seven people in the tournament. Is it always possible that the
seven people can be separated in two rooms such that none of the two rooms contains a cyclic
triplet?

8 Suppose circles W1 and W 2, with centres O1 and O2 respectively, intersect at points M and
N . Let the tangent on W2 at point N intersect W1 for the second time at B1 . Similarly, let the
tangent on W1 at point N intersect W2 for the second time at B2 . Let A1 be a point on W1 which
is on arc B1 N not containing M and suppose line A1 N intersects W2 at point A2 . Denote the
incentres of triangles B1 A1 N and B2 A2 N by I1 and I2 , respectively.*

B1
B2
M

O1
O2

A2
A1
N

Show that
∠I1 MI 2 = ∠O1 MO 2 .

*Given a triangle ABC, the incentre of the triangle is dened to be the intersection of the angle
bisectors of A, B, and C. To avoid cluttering, the incentre is omitted in the provided diagram.

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Note also that the diagram serves only as an aid and is not necessarily drawn to scale.

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Art of Problem Solving is an ACS WASC Accredited School.
AoPS Community 2010 Canadian Mathematical Olympiad Qualification
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Canadian Mathematical Olympiad Qualification Repechage 2010
www.artofproblemsolving.com/community/c4052
by TheMaskedMagician

loga x logb x
1 Suppose that a, b and x are positive real numbers. Prove that logab x = .
loga x + logb x

2 Two tangents AT and BT touch a circle at A and B, respectively, and meet perpendicularly at
T . Q is on AT , S is on BT , and R is on the circle, so that QRST is a rectangle with QT = 8 and
ST = 9. Determine the radius of the circle.

3 Prove that there is no real number x satisfying both equations

2x + 1 = 2 sin x
2x − 1 = 2 cos x.

4 Determine the smallest positive integer m with the property that m3 − 3m2 + 2m is divisible by
both 79 and 83.

5 The Fibonacci sequence is de ned by f1 = f2 = 1 and fn = fn−1 +fn−2 for n ≥ 3. A Pythagorean

triangle is a right-angled triangle with integer side lengths. Prove that f2k+1 is the hypotenuse
of a Pythagorean triangle for every positive integer k with k ≥ 2

6 There are 15 magazines on a table, and they cover the surface of the table entirely. Prove that
one can always take away 7 magazines in such a way that the remaining ones cover at least
8
of the area of the table surface
15

7 If (a, b, c) is a triple of real numbers, de fine

- g(a, b, c) = (a + b, b + c, a + c), and

- g n (a, b, c) = g(g n−1 (a, b, c)) for n ≥ 2
Suppose that there exists a positive integer n so that g n (a, b, c) = (a, b, c) for some (a, b, c) 6=
(0, 0, 0). Prove that g 6 (a, b, c) = (a, b, c)

8 Consider three parallelograms P1 , P2 , P3 . Parallelogram P3 is inside parallelogram P2 , and

the vertices of P3 are on the edges of P2 . Parallelogram P2 is inside parallelogram P1 , and the
vertices of P2 are on the edges of P1 . The sides of P3 are parallel to the sides of P1 . Prove that
one side of P3 has length at least half the length of the parallel side of P1 .

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Art of Problem Solving is an ACS WASC Accredited School.
AoPS Community 2009 Canadian Mathematical Olympiad Qualification
Repechage
Canadian Mathematical Olympiad Qualification Repechage 2009
www.artofproblemsolving.com/community/c4051
by TheMaskedMagician

1 Determine all solutions to the system of equations

x+y+z =2
x − y2 − z2 = 2
2

x − 3y 2 + z = 0

2 Triangle ABC is right-angled at C with AC = b and BC = a. If d is the length of the altitude

1 1 1
from C to AB, prove that 2 + 2 = 2
a b d

3 Prove that there does not exist a polynomial f (x) with integer coefficients for which f (2008) =
0 and f (2010) = 1867.

4 Three fair six-sided dice are thrown. Determine the probability that the sum of the numbers on
the three top faces is 6.

5 Determine all positive integers n for which n(n + 9) is a perfect square.

6 Triangle ABC is right-angled at C. AQ is drawn parallel to BC with Q and B on opposite

sides of AC so that when BQ is drawn, intersecting AC at P , we have P Q = 2AB. Prove that
∠ABC = 3∠P BC.

7 A rectangular sheet of paper is folded so that two diagonally opposite corners come together.
If the crease formed is the same length as the longer side of the sheet, what is the ratio of the
longer side of the sheet to the shorter side?

8 Determine an infinite family of quadruples (a, b, c, d) of positive integers, each of which is a

solution to a4 + b5 + c6 = d7 .

9 Suppose that m and k are positive integers. Determine the number of sequences x1 , x2 , x3 , . . . , xm−1 , xm
with

-xi an integer for i = 1, 2, 3, . . . , m,

-1 ≤ xi ≤ k for i = 1, 2, 3, . . . , m,
-x1 6= xm , and
-no two consecutive terms equal.

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10 Ten boxes are arranged in a circle. Each box initially contains a positive number of golf balls.
A move consists of taking all of the golf balls from one of the boxes and placing them into
the boxes that follow it in a counterclockwise direction, putting one ball into each box. Prove
that if the next move always starts with the box where the last ball of the previous move was
placed, then after some number of moves, we get back to the initial distribution of golf balls
in the boxes.

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Art of Problem Solving is an ACS WASC Accredited School.