kylemontgomery/cmo · Datasets at Hugging Face

id string	source string	problem string	solutions list
CMO-2002-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2002.pdf	Prove that for all positive real numbers \( a, b, \) and \( c \), \[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a + b + c, \] and determine when equality occurs.	[]
CMO-2002-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2002.pdf	Let \( \Gamma \) be a circle with radius \( r \). Let \( A \) and \( B \) be distinct points on \( \Gamma \) such that \( AB < \sqrt{3}r \). Let the circle with centre \( B \) and radius \( AB \) meet \( \Gamma \) again at \( C \). Let \( P \) be the point inside \( \Gamma \) such that triangle \( ABP \) is equilateral. Finally, let the line \( CP \) meet \( \Gamma \) again at \( Q \). Prove that \( PQ = r \).	[]
CMO-2002-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2002.pdf	Let \( N = \{0, 1, 2, \ldots\} \). Determine all functions \( f : N \to N \) such that \[ xf(y) + yf(x) = (x + y)f(x^2 + y^2) \] for all \( x \) and \( y \) in \( N \).	[]
CMO-2003-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2003.pdf	Consider a standard twelve-hour clock whose hour and minute hands move continuously. Let \( m \) be an integer, with \( 1 \leq m \leq 720 \). At precisely \( m \) minutes after 12:00, the angle made by the hour hand and minute hand is exactly \( 1^\circ \). Determine all possible values of \( m \).	[]
CMO-2003-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2003.pdf	Find the last three digits of the number \( 2003^{2002^{2001}} \).	[]
CMO-2003-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2003.pdf	Find all real positive solutions (if any) to \[ x^3 + y^3 + z^3 = x + y + z, \quad \text{and} \quad x^2 + y^2 + z^2 = xyz. \]	[]
CMO-2003-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2003.pdf	Prove that when three circles share the same chord \( AB \), every line through \( A \) different from \( AB \) determines the same ratio \( XY : YZ \), where \( X \) is an arbitrary point different from \( B \) on the first circle while \( Y \) and \( Z \) are the points where \( AX \) intersects the other two circles (labelled so that \( Y \) is between \( X \) and \( Z \)).	[]
CMO-2003-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2003.pdf	Let \( S \) be a set of \( n \) points in the plane such that any two points of \( S \) are at least 1 unit apart. Prove there is a subset \( T \) of \( S \) with at least \( n/7 \) points such that any two points of \( T \) are at least \( \sqrt{3} \) units apart.	[]
CMO-2004-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2004.pdf	Find all ordered triples \( (x, y, z) \) of real numbers which satisfy the following system of equations: \[ \begin{cases} xy = z - x - y xz = y - x - z yz = x - y - z \end{cases} \]	[]
CMO-2004-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2004.pdf	How many ways can 8 mutually non-attacking rooks be placed on the \( 9 \times 9 \) chessboard (shown here) so that all 8 rooks are on squares of the same colour? [Two rooks are said to be attacking each other if they are placed in the same row or column of the board.]	[]
CMO-2004-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2004.pdf	Let \( A, B, C, D \) be four points on a circle (occurring in clockwise order), with \( AB < AD \) and \( BC > CD \). Let the bisector of angle \( BAD \) meet the circle at \( X \) and the bisector of angle \( BCD \) meet the circle at \( Y \). Consider the hexagon formed by these six points on the circle. If four of the six sides of the hexagon have equal length, prove that \( BD \) must be a diameter of the circle.	[]
CMO-2004-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2004.pdf	Let \( p \) be an odd prime. Prove that \[ \sum_{k=1}^{p-1} k^{2p-1} \equiv \frac{p(p+1)}{2} \pmod{p^2}. \] [Note that \( a \equiv b \pmod{m} \) means that \( a - b \) is divisible by \( m \).]	[]
CMO-2004-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2004.pdf	Let \( T \) be the set of all positive integer divisors of \( 2004^{100} \). What is the largest possible number of elements that a subset \( S \) of \( T \) can have if no element of \( S \) is an integer multiple of any other element of \( S \)?	[]
CMO-2005-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2005.pdf	Consider an equilateral triangle of side length \( n \), which is divided into unit triangles, as shown. Let \( f(n) \) be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in our path share a common edge and the path never travels up (from a lower row to a higher row) or revisits a triangle. An example of one such path is illustrated below for \( n = 5 \). Determine the value of \( f(2005) \).	[]
CMO-2005-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2005.pdf	Let \( (a, b, c) \) be a Pythagorean triple, i.e., a triplet of positive integers with \( a^2 + b^2 = c^2 \). \begin{itemize} \item[(a)] Prove that \( (c/a + c/b)^2 > 8. \) \item[(b)] Prove that there does not exist any integer \( n \) for which we can find a Pythagorean triple \( (a, b, c) \) satisfying \( (c/a + c/b)^2 = n. \) \end{itemize}	[]
CMO-2005-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2005.pdf	Let \( S \) be a set of \( n \geq 3 \) points in the interior of a circle. \begin{itemize} \item[(a)] Show that there are three distinct points \( a, b, c \in S \) and three distinct points \( A, B, C \) on the circle such that \( a \) is (strictly) closer to \( A \) than any other point in \( S \), \( b \) is closer to \( B \) than any other point in \( S \) and \( c \) is closer to \( C \) than any other point in \( S \). \item[(b)] Show that for no value of \( n \) can four such points in \( S \) (and corresponding points on the circle) be guaranteed. \end{itemize}	[]
CMO-2005-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2005.pdf	Let \( ABC \) be a triangle with circumradius \( R \), perimeter \( P \) and area \( K \). Determine the maximum value of \( KP/R^3 \).	[]
CMO-2005-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2005.pdf	Let’s say that an ordered triple of positive integers \( (a, b, c) \) is \( n \)-powerful if \( a \leq b \leq c \), \( \gcd(a, b, c) = 1 \), and \( a^n + b^n + c^n \) is divisible by \( a + b + c \). For example, \( (1, 2, 2) \) is 5-powerful. \begin{itemize} \item[(a)] Determine all ordered triples (if any) which are \( n \)-powerful for all \( n \geq 1 \). \item[(b)] Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful. \end{itemize} [Note that \( \gcd(a, b, c) \) is the greatest common divisor of \( a, b \) and \( c \).]	[]
CMO-2006-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2006.pdf	Let \( f(n, k) \) be the number of ways of distributing \( k \) candies to \( n \) children so that each child receives at most 2 candies. For example, if \( n = 3 \), then \( f(3, 7) = 0 \), \( f(3, 6) = 1 \) and \( f(3, 4) = 6 \). Determine the value of \[ f(2006, 1) + f(2006, 4) + f(2006, 7) + \cdots + f(2006, 1000) + f(2006, 1003). \]	[]
CMO-2006-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2006.pdf	Let \( ABC \) be an acute-angled triangle. Inscribe a rectangle \( DEFG \) in this triangle so that \( D \) is on \( AB \), \( E \) is on \( AC \) and both \( F \) and \( G \) are on \( BC \). Describe the locus of (i.e., the curve occupied by) the intersections of the diagonals of all possible rectangles \( DEFG \).	[]
CMO-2006-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2006.pdf	In a rectangular array of nonnegative real numbers with \( m \) rows and \( n \) columns, each row and each column contains at least one positive element. Moreover, if a row and a column intersect in a positive element, then the sums of their elements are the same. Prove that \( m = n \).	[]
CMO-2006-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2006.pdf	Consider a round-robin tournament with \( 2n + 1 \) teams, where each team plays each other team exactly once. We say that three teams \( X \), \( Y \) and \( Z \), form a cycle triplet if \( X \) beats \( Y \), \( Y \) beats \( Z \), and \( Z \) beats \( X \). There are no ties. (a) Determine the minimum number of cycle triplets possible. (b) Determine the maximum number of cycle triplets possible.	[]
CMO-2006-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2006.pdf	The vertices of a right triangle \( ABC \) inscribed in a circle divide the circumference into three arcs. The right angle is at \( A \), so that the opposite arc \( BC \) is a semicircle while arc \( AB \) and arc \( AC \) are supplementary. To each of the three arcs, we draw a tangent such that its point of tangency is the midpoint of that portion of the tangent intercepted by the extended lines \( AB \) and \( AC \). More precisely, the point \( D \) on arc \( BC \) is the midpoint of the segment joining the points \( D' \) and \( D'' \) where the tangent at \( D \) intersects the extended lines \( AB \) and \( AC \). Similarly for \( E \) on arc \( AC \) and \( F \) on arc \( AB \). Prove that triangle \( DEF \) is equilateral.	[]
CMO-2007-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2007.pdf	What is the maximum number of non-overlapping \(2 \times 1\) dominoes that can be placed on a \(8 \times 9\) checkerboard if six of them are placed as shown? Each domino must be placed horizontally or vertically so as to cover two adjacent squares of the board. (A diagram shows six dominoes already placed on a checkerboard.)	[]
CMO-2007-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2007.pdf	You are given a pair of triangles for which (a) two sides of one triangle are equal in length to two sides of the second triangle, and (b) the triangles are similar, but not necessarily congruent. Prove that the ratio of the sides that correspond under the similarity is a number between \( \tfrac{1}{2}(\sqrt{5} - 1) \) and \( \tfrac{1}{2}(\sqrt{5} + 1) \).	[]
CMO-2007-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2007.pdf	Suppose that \( f \) is a real-valued function for which \[ f(xy) + f(y - x) \geq f(y + x) \] for all real numbers \( x \) and \( y \). (a) Give a nonconstant polynomial that satisfies the condition. (b) Prove that \( f(x) \geq 0 \) for all real \( x \).	[]
CMO-2007-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2007.pdf	For two real numbers \( a, b \), with \( ab eq 1 \), define the \( * \) operation by \[ a * b = \frac{a + b - 2ab}{1 - ab}. \] Start with a list of \( n \geq 2 \) real numbers whose entries \( x \) all satisfy \( 0 < x < 1 \). Select any two numbers \( a \) and \( b \) in the list; remove them and put the number \( a * b \) at the end of the list, thereby reducing its length by one. Repeat this procedure until a single number remains. (a) Prove that this single number is the same regardless of the choice of pair at each stage. (b) Suppose that the condition on the numbers \( x \) in \( S \) is weakened to \( 0 < x \leq 1 \). What happens if \( S \) contains exactly one 1?	[]
CMO-2007-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2007.pdf	Let the incircle of triangle \( ABC \) touch sides \( BC, CA \) and \( AB \) at \( D, E \) and \( F \), respectively. Let \( \Gamma, \Gamma_1, \Gamma_2 \) and \( \Gamma_3 \) denote the circumcircles of triangle \( ABC, AEF, BDF \) and \( CDE \) respectively. Let \( \Gamma \) and \( \Gamma_1 \) intersect at \( A \) and \( P \), \( \Gamma \) and \( \Gamma_2 \) intersect at \( B \) and \( Q \), and \( \Gamma \) and \( \Gamma_3 \) intersect at \( C \) and \( R \). (a) Prove that the circles \( \Gamma_1, \Gamma_2 \) and \( \Gamma_3 \) intersect in a common point. (b) Show that \( PD, QE \) and \( RF \) are concurrent.	[]
CMO-2008-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2008.pdf	Let \(ABCD\) be a convex quadrilateral for which \(AB\) is the longest side. Points \(M\) and \(N\) are located on sides \(AB\) and \(BC\) respectively, so that each of the segments \(AN\) and \(CM\) divides the quadrilateral into two parts of equal area. Prove that the segment \(MN\) bisects the diagonal \(BD\).	[]
CMO-2008-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2008.pdf	Determine all functions \(f\) defined on the set of rational numbers that take rational values for which \[ f(2f(x) + f(y)) = 2x + y , \] for each \(x\) and \(y\).	[]
CMO-2008-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2008.pdf	Let \(a, b, c\) be positive real numbers for which \(a + b + c = 1\). Prove that \[ \frac{a - bc}{a + bc} + \frac{b - ca}{b + ca} + \frac{c - ab}{c + ab} \leq \frac{3}{2} . \]	[]
CMO-2008-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2008.pdf	Determine all functions \(f\) defined on the natural numbers that take values among the natural numbers for which \[ (f(n))^p \equiv n \pmod{f(p)} \] for all \(n \in \mathbb{N}\) and all prime numbers \(p\).	[]
CMO-2008-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2008.pdf	A \emph{self-avoiding rook walk} on a chessboard (a rectangular grid of unit squares) is a path traced by a sequence of moves parallel to an edge of the board from one unit square to another, such that each begins where the previous move ended and such that no move ever crosses a square that has previously been crossed, \emph{i.e.}, the rook’s path is non-self-intersecting. Let \(R(m,n)\) be the number of self-avoiding rook walks on an \(m \times n\) (\(m\) rows, \(n\) columns) chessboard which begin at the lower-left corner and end at the upper-left corner. For example, \(R(m,1) = 1\) for all natural numbers \(m\); \(R(2,2) = 2\); \(R(3,2) = 4\); \(R(3,3) = 11\). Find a formula for \(R(3,n)\) for each natural number \(n\).	[]
CMO-2009-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2009.pdf	Given an \( m \times n \) grid with squares coloured either black or white, we say that a black square in the grid is \emph{stranded} if there is some square to its left in the same row that is white and there is some square above it in the same column that is white (see Figure 1). [Figure shows a 4 × 5 grid with no stranded black squares.] Find a closed formula for the number of \( 2 \times n \) grids with no stranded black squares.	[]
CMO-2009-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2009.pdf	Two circles of different radii are cut out of cardboard. Each circle is subdivided into 200 equal sectors. On each circle 100 sectors are painted white and the other 100 are painted black. The smaller circle is then placed on top of the larger circle, so that their centers coincide. Show that one can rotate the small circle so that the sectors on the two circles line up and at least 100 sectors on the small circle lie over sectors of the same color on the big circle.	[]
CMO-2009-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2009.pdf	Define \[ f(x, y, z) = \frac{(xy + yz + zx)(x + y + z)}{(x + y)(x + z)(y + z)}. \] Determine the set of real numbers \( r \) for which there exists a triplet \( (x, y, z) \) of positive real numbers satisfying \( f(x, y, z) = r \).	[]
CMO-2009-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2009.pdf	Find all ordered pairs \( (a, b) \) such that \( a \) and \( b \) are integers and \( 3^a + 7^b \) is a perfect square.	[]
CMO-2009-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2009.pdf	A set of points is marked on the plane, with the property that any three marked points can be covered with a disk of radius 1. Prove that the set of all marked points can be covered with a disk of radius 1.	[]
CMO-2010-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2010.pdf	For a positive integer \( n \), an \( n \)-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to \( n \) squares in the \( n \)th row, such that all the left-most squares in each row are aligned vertically. For example, the 5-staircase is shown below. (A diagram of a 5-staircase is shown, with rows of 1 to 5 unit squares aligned on the left.) Let \( f(n) \) denote the minimum number of square tiles required to tile the \( n \)-staircase, where the side lengths of the square tiles can be any positive integer. For example, \( f(2) = 3 \) and \( f(4) = 7 \). (a) Find all \( n \) such that \( f(n) = n \). (b) Find all \( n \) such that \( f(n) = n + 1 \).	[]
CMO-2010-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2010.pdf	Let \( A, B, P \) be three points on a circle. Prove that if \( a \) and \( b \) are the distances from \( P \) to the tangents at \( A \) and \( B \) and \( c \) is the distance from \( P \) to the chord \( AB \), then \( c^2 = ab \).	[]
CMO-2010-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2010.pdf	Three speed skaters have a friendly “race” on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does 1 lap a minute, the fastest one does 3.14 laps a minute, and the middle one does \( L \) laps a minute for some \( 1 < L < 3.14 \). The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting point.) Find how many different choices for \( L \) are there such that exactly 117 passings occur before the end of the race. (A passing is defined when one skater passes another one. The beginning and the end of the race when all three skaters are together are not counted as passings.)	[]
CMO-2010-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2010.pdf	Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex \( P \) and change the colour of \( P \) and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a sequence of operations of this type? (A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex A and vertex B, then B is called a neighbour of A.)	[]
CMO-2010-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2010.pdf	Let \( P(x) \) and \( Q(x) \) be polynomials with integer coefficients. Let \( a_n = n! + n \). Show that if \( P(a_n)/Q(a_n) \) is an integer for every \( n \), then \( P(n)/Q(n) \) is an integer for every integer \( n \) such that \( Q(n) e 0 \).	[]
CMO-2011-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2011.pdf	Consider 70-digit numbers \( n \), with the property that each of the digits 1, 2, 3, \ldots, 7 appears in the decimal expansion of \( n \) ten times (and 8, 9, and 0 do not appear). Show that no number of this form can divide another number of this form.	[]
CMO-2011-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2011.pdf	Let \( ABCD \) be a cyclic quadrilateral whose opposite sides are not parallel, \( X \) the intersection of \( AB \) and \( CD \), and \( Y \) the intersection of \( AD \) and \( BC \). Let the angle bisector of \( \angle X D \) intersect \( AD, BC \) at \( E, F \) respectively and let the angle bisector of \( \angle Y B \) intersect \( AB, CD \) at \( G, H \) respectively. Prove that \( EGFH \) is a parallelogram.	[]
CMO-2011-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2011.pdf	Amy has divided a square up into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates \( x \), the sum of these numbers. If the total area of the white rectangles equals the total area of the red rectangles, what is the smallest possible value of \( x \)?	[]
CMO-2011-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2011.pdf	Show that there exists a positive integer \( N \) such that for all integers \( a > N \), there exists a contiguous substring of the decimal expansion of \( a \) that is divisible by 2011. (For instance, if \( a = 153204 \), then 15, 532, and 0 are all contiguous substrings of \( a \). Note that 0 is divisible by 2011.)	[]
CMO-2011-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2011.pdf	Let \( d \) be a positive integer. Show that for every integer \( S \), there exists an integer \( n > 0 \) and a sequence \( \epsilon_1, \epsilon_2, \ldots, \epsilon_n \), where for any \( k \), \( \epsilon_k = 1 \) or \( \epsilon_k = -1 \), such that \[ S = \epsilon_1(1 + d)^2 + \epsilon_2(1 + 2d)^2 + \epsilon_3(1 + 3d)^2 + \cdots + \epsilon_n(1 + nd)^2. \]	[]
CMO-2012-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2012.pdf	Let \( x, y \) and \( z \) be positive real numbers. Show that \[ x^2 + xy^2 + xyz^2 \geq 4xyz - 4. \]	[]
CMO-2012-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2012.pdf	For any positive integers \( n \) and \( k \), let \( L(n,k) \) be the least common multiple of the \( k \) consecutive integers \( n, n+1, \ldots, n+k-1 \). Show that for any integer \( b \), there exist integers \( n \) and \( k \) such that \[ L(n,k) > b \cdot L(n+1,k). \]	[]
CMO-2012-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2012.pdf	Let \( ABCD \) be a convex quadrilateral and let \( P \) be the point of intersection of \( AC \) and \( BD \). Suppose that \[ AC + AD = BC + BD. \] Prove that the internal angle bisectors of \( \angle ACB \), \( \angle ADB \), and \( \angle APB \) meet at a common point.	[]
CMO-2012-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2012.pdf	A number of robots are placed on the squares of a finite, rectangular grid of squares. A square can hold any number of robots. Every edge of the grid is classified as either passable or impassable. All edges on the boundary of the grid are impassable. You can give any of the commands \textit{up}, \textit{down}, \textit{left}, or \textit{right}. All of the robots then simultaneously try to move in the specified direction. If the edge adjacent to a robot in that direction is passable, the robot moves across the edge and into the next square. Otherwise, the robot remains on its current square. You can then give another command of \textit{up}, \textit{down}, \textit{left}, or \textit{right}, then another, for as long as you want. Suppose that for any individual robot, and any square on the grid, there is a finite sequence of commands that will move that robot to that square. Prove that you can also give a finite sequence of commands such that \textbf{all} of the robots end up on the same square at the same time.	[]
CMO-2012-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2012.pdf	A bookshelf contains \( n \) volumes, labelled \( 1 \) to \( n \) in some order. The librarian wishes to put them in the correct order as follows. The librarian selects a volume that is too far to the right, say the volume with label \( k \), takes it out, and inserts it so that it is in the \( k \)-th place. For example, if the bookshelf contains the volumes 1, 3, 2, 4 in that order, the librarian could take out volume 2 and place it in the second position. The books will then be in the correct order 1, 2, 3, 4. (a) Show that if this process is repeated, then, however the librarian makes the selections, all the volumes will eventually be in the correct order. (b) What is the largest number of steps that this process can take?	[]
CMO-2013-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2013.pdf	Determine all polynomials \( P(x) \) with real coefficients such that \[ (x+1)P(x-1) - (x-1)P(x) \] is a constant polynomial.	[]
CMO-2013-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2013.pdf	The sequence \( a_1, a_2, \ldots, a_n \) consists of the numbers \( 1, 2, \ldots, n \) in some order. For which positive integers \( n \) is it possible that the \( n+1 \) numbers \( 0, a_1, a_1+a_2, a_1+a_2+a_3, \ldots, a_1 + a_2 + \cdots + a_n \) all have different remainders when divided by \( n+1 \)?	[]
CMO-2013-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2013.pdf	Let \( G \) be the centroid of a right-angled triangle \( ABC \) with \( \angle BCA = 90^\circ \). Let \( P \) be the point on ray \( AG \) such that \( \angle CPA = \angle CAB \), and let \( Q \) be the point on ray \( BG \) such that \( \angle CQB = \angle ABC \). Prove that the circumcircles of triangles \( AQG \) and \( BPG \) meet at a point on side \( AB \).	[]
CMO-2013-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2013.pdf	Let \( n \) be a positive integer. For any positive integer \( j \) and positive real number \( r \), define \( f_j(r) \) and \( g_j(r) \) by \[ f_j(r) = \min(jr, n) + \min\left( \frac{j}{r}, n \right), \quad \text{and} \quad g_j(r) = \min(\lfloor jr \rfloor, n) + \min\left( \left\lceil \frac{j}{r} \right\rceil, n \right), \] where \( \lceil x \rceil \) denotes the smallest integer greater than or equal to \( x \). Prove that \[ \sum_{j=1}^n f_j(r) \leq n^2 + n \leq \sum_{j=1}^n g_j(r) \] for all positive real numbers \( r \).	[]
CMO-2013-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2013.pdf	Let \( O \) denote the circumcentre of an acute-angled triangle \( ABC \). Let point \( P \) on side \( AB \) be such that \( \angle BOP = \angle ABC \), and let point \( Q \) on side \( AC \) be such that \( \angle COQ = \angle ACB \). Prove that the reflection of \( BC \) in the line \( PQ \) is tangent to the circumcircle of triangle \( APQ \).	[]
CMO-2014-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2014.pdf	Let \( a_1, a_2, \ldots, a_n \) be positive real numbers whose product is 1. Show that the sum \[ \frac{a_1}{1 + a_1} + \frac{a_2}{(1 + a_1)(1 + a_2)} + \frac{a_3}{(1 + a_1)(1 + a_2)(1 + a_3)} + \cdots + \frac{a_n}{(1 + a_1)(1 + a_2) \cdots (1 + a_n)} \] is greater than or equal to \[ \frac{2^n - 1}{2^n}. \]	[]
CMO-2014-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2014.pdf	Let \( m \) and \( n \) be odd positive integers. Each square of an \( m \) by \( n \) board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of \( m \) and \( n \).	[]
CMO-2014-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2014.pdf	Let \( p \) be a fixed odd prime. A \( p \)-tuple \( (a_1, a_2, a_3, \ldots, a_p) \) of integers is said to be good if (i) \( 0 \leq a_i \leq p - 1 \) for all \( i \), and (ii) \( a_1 + a_2 + a_3 + \cdots + a_p \) is not divisible by \( p \), and (iii) \( a_1a_2 + a_2a_3 + a_3a_4 + \cdots + a_pa_1 \) is divisible by \( p \). Determine the number of good \( p \)-tuples.	[]
CMO-2014-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2014.pdf	The quadrilateral \( ABCD \) is inscribed in a circle. The point \( P \) lies in the interior of \( ABCD \), and \( \angle PAB = \angle PBC = \angle PCD = \angle PDA \). The lines \( AD \) and \( BC \) meet at \( Q \), and the lines \( AB \) and \( CD \) meet at \( R \). Prove that the lines \( PQ \) and \( PR \) form the same angle as the diagonals of \( ABCD \).	[]
CMO-2014-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2014.pdf	Fix positive integers \( n \) and \( k \geq 2 \). A list of \( n \) integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add 1 to all of them or subtract 1 from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least \( n - k + 2 \) of the numbers on the blackboard are all simultaneously divisible by \( k \).	[]
CMO-2015-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2015.pdf	Let \( \mathbb{N} = \{1, 2, 3, \ldots\} \) be the set of positive integers. Find all functions \( f \), defined on \( \mathbb{N} \) and taking values in \( \mathbb{N} \), such that \[ (n - 1)^2 < f(n)f(f(n)) < n^2 + n \] for every positive integer \( n \).	[]
CMO-2015-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2015.pdf	Let \( ABC \) be an acute-angled triangle with altitudes \( AD, BE, \) and \( CF \). Let \( H \) be the orthocentre, that is, the point where the altitudes meet. Prove that \[ rac{AB \cdot AC + BC \cdot BA + CA \cdot CB}{AH \cdot AD + BH \cdot BE + CH \cdot CF} \leq 2. \]	[]
CMO-2015-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2015.pdf	On a \( (4n + 2) imes (4n + 2) \) square grid, a turtle can move between squares sharing a side. The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of \( n \), what is the largest positive integer \( k \) such that there must be a row or column that the turtle has entered at least \( k \) distinct times?	[]
CMO-2015-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2015.pdf	Let \( ABC \) be an acute-angled triangle with circumcenter \( O \). Let \( \Gamma \) be a circle with centre on the altitude from \( A \) in \( ABC \), passing through vertex \( A \) and points \( P \) and \( Q \) on sides \( AB \) and \( AC \). Assume that \( BP \cdot CQ = AP \cdot AQ \). Prove that \( \Gamma \) is tangent to the circumcircle of triangle \( BOC \).	[]
CMO-2015-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2015.pdf	Let \( p \) be a prime number for which \( rac{p - 1}{2} \) is also prime, and let \( a, b, c \) be integers not divisible by \( p \). Prove that there are at most \( 1 + \sqrt{2p} \) positive integers \( n \) such that \( n < p \) and \( p \) divides \( a^n + b^n + c^n \).	[]
CMO-2016-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2016.pdf	The integers \(1, 2, 3, \ldots, 2016\) are written on a board. You can choose any two numbers on the board and replace them with their average. For example, you can replace 1 and 2 with 1.5, or you can replace 1 and 3 with a second copy of 2. After 2015 replacements of this kind, the board will have only one number left on it. (a) Prove that there is a sequence of replacements that will make the final number equal to 2. (b) Prove that there is a sequence of replacements that will make the final number equal to 1000.	[]
CMO-2016-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2016.pdf	Consider the following system of 10 equations in 10 real variables \(v_1, \ldots, v_{10}\): \[ v_i = 1 + \frac{6v_i^2}{v_1^2 + v_2^2 + \cdots + v_{10}^2} \quad (i = 1, \ldots, 10). \] Find all 10-tuples \((v_1, v_2, \ldots, v_{10})\) that are solutions of this system.	[]
CMO-2016-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2016.pdf	Find all polynomials \(P(x)\) with integer coefficients such that \(P(P(n) + n)\) is a prime number for infinitely many integers \(n\).	[]
CMO-2016-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2016.pdf	Let \(A, B,\) and \(F\) be positive integers, and assume \(A < B < 2A\). A flea is at the number 0 on the number line. The flea can move by jumping to the right by \(A\) or by \(B\). Before the flea starts jumping, Lavaman chooses finitely many intervals \(\{m+1, m+2, \ldots, m+A\}\) consisting of \(A\) consecutive positive integers, and places lava at all of the integers in the intervals. The intervals must be chosen so that: (i) any two distinct intervals are disjoint and not adjacent; (ii) there are at least \(F\) positive integers with no lava between any two intervals; and (iii) no lava is placed at any integer less than \(F\). Prove that the smallest \(F\) for which the flea can jump over all the intervals and avoid all the lava, regardless of what Lavaman does, is \[ F = (n - 1)A + B, \] where \(n\) is the positive integer such that \[ \frac{A}{n+1} \leq B - A < \frac{A}{n}. \]	[]
CMO-2016-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2016.pdf	Let \(\triangle ABC\) be an acute-angled triangle with altitudes \(AD\) and \(BE\) meeting at \(H\). Let \(M\) be the midpoint of segment \(AB\), and suppose that the circumcircles of \(\triangle DEM\) and \(\triangle ABH\) meet at points \(P\) and \(Q\) with \(P\) on the same side of \(CH\) as \(A\). Prove that the lines \(ED\), \(PH\), and \(MQ\) all pass through a single point on the circumcircle of \(\triangle ABC\).	[]
CMO-2017-1	https://cms.math.ca/wp-content/uploads/2024/03/exam2017.pdf	Let \( a \), \( b \), and \( c \) be non-negative real numbers, no two of which are equal. Prove that \[ \frac{a^2}{(b - c)^2} + \frac{b^2}{(c - a)^2} + \frac{c^2}{(a - b)^2} > 2. \]	[]
CMO-2017-2	https://cms.math.ca/wp-content/uploads/2024/03/exam2017.pdf	Let \( f \) be a function from the set of positive integers to itself such that, for every \( n \), the number of positive integer divisors of \( n \) is equal to \( f(f(n)) \). For example, \( f(f(6)) = 4 \) and \( f(f(25)) = 3 \). Prove that if \( p \) is prime then \( f(p) \) is also prime.	[]
CMO-2017-3	https://cms.math.ca/wp-content/uploads/2024/03/exam2017.pdf	Let \( n \) be a positive integer, and define \( S_n = \{1, 2, \ldots, n\} \). Consider a non-empty subset \( T \) of \( S_n \). We say that \( T \) is balanced if the median of \( T \) is equal to the average of \( T \). For example, for \( n = 9 \), each of the subsets \( \{7\}, \{2, 5\}, \{2, 3, 4\}, \{5, 6, 8, 9\}, \{1, 4, 5, 7, 8\} \) is balanced; however, the subsets \( \{2, 4, 5\} \) and \( \{1, 2, 3, 5\} \) are not balanced. For each \( n \geq 1 \), prove that the number of balanced subsets of \( S_n \) is odd. (To define the median of a set of \( k \) numbers, first put the numbers in increasing order; then the median is the middle number if \( k \) is odd, and the average of the two middle numbers if \( k \) is even. For example, the median of \( \{1, 3, 4, 8, 9\} \) is 4, and the median of \( \{1, 3, 4, 7, 8, 9\} \) is \( (4 + 7)/2 = 5.5 \).)	[]
CMO-2017-4	https://cms.math.ca/wp-content/uploads/2024/03/exam2017.pdf	Points \( P \) and \( Q \) lie inside parallelogram \( ABCD \) and are such that triangles \( ABP \) and \( BCQ \) are equilateral. Prove that the line through \( P \) perpendicular to \( DP \) and the line through \( Q \) perpendicular to \( DQ \) meet on the altitude from \( B \) in triangle \( ABC \).	[]
CMO-2017-5	https://cms.math.ca/wp-content/uploads/2024/03/exam2017.pdf	One hundred circles of radius one are positioned in the plane so that the area of any triangle formed by the centres of three of these circles is at most 2017. Prove that there is a line intersecting at least three of these circles.	[]
CMO-2018-1	https://cms.math.ca/wp-content/uploads/2024/03/exam2018.pdf	Consider an arrangement of tokens in the plane, not necessarily at distinct points. We are allowed to apply a sequence of moves of the following kind: Select a pair of tokens at points \( A \) and \( B \) and move both of them to the midpoint of \( A \) and \( B \). We say that an arrangement of \( n \) tokens is \emph{collapsible} if it is possible to end up with all \( n \) tokens at the same point after a finite number of moves. Prove that every arrangement of \( n \) tokens is collapsible if and only if \( n \) is a power of 2.	[]
CMO-2018-2	https://cms.math.ca/wp-content/uploads/2024/03/exam2018.pdf	Let five points on a circle be labelled \( A, B, C, D, \) and \( E \) in clockwise order. Assume \( AE = DE \) and let \( P \) be the intersection of \( AC \) and \( BD \). Let \( Q \) be the point on the line through \( A \) and \( B \) such that \( A \) is between \( B \) and \( Q \) and \( AQ = DP \). Similarly, let \( R \) be the point on the line through \( C \) and \( D \) such that \( D \) is between \( C \) and \( R \) and \( DR = AP \). Prove that \( PE \) is perpendicular to \( QR \).	[]
CMO-2018-3	https://cms.math.ca/wp-content/uploads/2024/03/exam2018.pdf	Two positive integers \( a \) and \( b \) are \emph{prime-related} if \( a = pb \) or \( b = pa \) for some prime \( p \). Find all positive integers \( n \), such that \( n \) has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related. Note that 1 and \( n \) are included as divisors.	[]
CMO-2018-4	https://cms.math.ca/wp-content/uploads/2024/03/exam2018.pdf	Find all polynomials \( p(x) \) with real coefficients that have the following property: There exists a polynomial \( q(x) \) with real coefficients such that \[ p(1) + p(2) + p(3) + \cdots + p(n) = p(n)q(n) \] for all positive integers \( n \).	[]
CMO-2018-5	https://cms.math.ca/wp-content/uploads/2024/03/exam2018.pdf	Let \( k \) be a given even positive integer. Sarah first picks a positive integer \( N \) greater than 1 and proceeds to alter it as follows: every minute, she chooses a prime divisor \( p \) of the current value of \( N \), and multiplies the current \( N \) by \( p^k - p^{-1} \) to produce the next value of \( N \). Prove that there are infinitely many even positive integers \( k \) such that, no matter what choices Sarah makes, her number \( N \) will at some point be divisible by 2018.	[]
CMO-2019-1	https://cms.math.ca/wp-content/uploads/2024/03/exam2019.pdf	Amy has drawn three points in a plane, \( A, B, \) and \( C \), such that \( AB = BC = CA = 6 \). Amy is allowed to draw a new point if it is the circumcenter of a triangle whose vertices she has already drawn. For example, she can draw the circumcenter \( O \) of triangle \( ABC \), and then afterwards she can draw the circumcenter of triangle \( ABO \). (a) Prove that Amy can eventually draw a point whose distance from a previously drawn point is greater than 7. (b) Prove that Amy can eventually draw a point whose distance from a previously drawn point is greater than 2019. (Recall that the circumcenter of a triangle is the center of the circle that passes through its three vertices.)	[]
CMO-2019-2	https://cms.math.ca/wp-content/uploads/2024/03/exam2019.pdf	Let \( a \) and \( b \) be positive integers such that \( a + b^3 \) is divisible by \( a^2 + 3ab + 3b^2 - 1 \). Prove that \( a^2 + 3ab + 3b^2 - 1 \) is divisible by the cube of an integer greater than 1.	[]
CMO-2019-3	https://cms.math.ca/wp-content/uploads/2024/03/exam2019.pdf	Let \( m \) and \( n \) be positive integers. A \( 2m \times 2n \) grid of squares is coloured in the usual chessboard fashion. Find the number of ways of placing \( mn \) counters on the white squares, at most one counter per square, so that no two counters are on white squares that are diagonally adjacent. An example of a way to place the counters when \( m = 2 \) and \( n = 3 \) is shown below. [Figure shows a 4x6 grid with black and white squares in chessboard pattern, and 6 counters placed on white squares such that no two are diagonally adjacent.]	[]
CMO-2019-4	https://cms.math.ca/wp-content/uploads/2024/03/exam2019.pdf	Let \( n \) be an integer greater than 1, and let \( a_0, a_1, \ldots, a_n \) be real numbers with \( a_1 = a_{n-1} = 0 \). Prove that for any real number \( k \), \[ \|a_0\| - \|a_n\| \leq \sum_{i=0}^{n-2} \|a_i - k a_{i+1} - a_{i+2}\|. \]	[]
CMO-2019-5	https://cms.math.ca/wp-content/uploads/2024/03/exam2019.pdf	David and Jacob are playing a game of connecting \( n \geq 3 \) points drawn in a plane. No three of the points are collinear. On each player's turn, he chooses two points to connect by a new line segment. The first player to complete a cycle consisting of an odd number of line segments loses the game. (Both endpoints of each line segment in the cycle must be among the \( n \) given points, not points which arise later as intersections of segments.) Assuming David goes first, determine all \( n \) for which he has a winning strategy.	[]
CMO-2020-1	https://cms.math.ca/wp-content/uploads/2024/03/exam2020.pdf	Let \( S \) be a set of \( n \geq 3 \) positive real numbers. Show that the largest possible number of distinct integer powers of three that can be written as the sum of three distinct elements of \( S \) is \( n - 2 \).	[]
CMO-2020-2	https://cms.math.ca/wp-content/uploads/2024/03/exam2020.pdf	A circle is inscribed in a rhombus \( ABCD \). Points \( P \) and \( Q \) vary on line segments \( AB \) and \( AD \), respectively, so that \( PQ \) is tangent to the circle. Show that for all such line segments \( PQ \), the area of triangle \( CPQ \) is constant. (Figure shows rhombus \(ABCD\) with an inscribed circle. Points \(P\) on \(AB\) and \(Q\) on \(AD\) are such that line \(PQ\) is tangent to the circle. Triangle \(CPQ\) is formed.)	[]
CMO-2020-3	https://cms.math.ca/wp-content/uploads/2024/03/exam2020.pdf	A purse contains a finite number of coins, each with distinct positive integer values. Is it possible that there are exactly 2020 ways to use coins from the purse to make the value 2020?	[]
CMO-2020-4	https://cms.math.ca/wp-content/uploads/2024/03/exam2020.pdf	Let \( S = \{1, 4, 8, 9, 16, \ldots\} \) be the set of perfect powers of integers, i.e. numbers of the form \( n^k \) where \( n, k \) are positive integers and \( k \geq 2 \). Write \( S = \{a_1, a_2, a_3, \ldots\} \) with terms in increasing order, so that \( a_1 < a_2 < a_3 < \cdots \). Prove that there exist infinitely many integers \( m \) such that 9999 divides the difference \( a_{m+1} - a_m \).	[]
CMO-2020-5	https://cms.math.ca/wp-content/uploads/2024/03/exam2020.pdf	There are 19,998 people on a social media platform, where any pair of them may or may not be \emph{friends}. For any group of 9,999 people, there are at least 9,999 pairs of them that are friends. What is the least number of friendships, that is, the least number of pairs of people that are friends, that must be among the 19,998 people?	[]
CMO-2021-1	https://cms.math.ca/wp-content/uploads/2021/04/CMO-2021-questions-en-4.pdf	Let \(ABCD\) be a trapezoid with \(AB\) parallel to \(CD\), \(\|AB\| > \|CD\|\), and equal edges \(\|AD\| = \|BC\|\). Let \(I\) be the center of the circle tangent to lines \(AB\), \(AC\) and \(BD\), where \(A\) and \(I\) are on opposite sides of \(BD\). Let \(J\) be the center of the circle tangent to lines \(CD\), \(AC\) and \(BD\), where \(D\) and \(J\) are on opposite sides of \(AC\). Prove that \(\|CI\| = \|JB\|\).	[]
CMO-2021-2	https://cms.math.ca/wp-content/uploads/2021/04/CMO-2021-questions-en-4.pdf	Let \(n \geq 2\) be some fixed positive integer and suppose that \(a_1, a_2, \ldots, a_n\) are positive real numbers satisfying \(a_1 + a_2 + \cdots + a_n = 2^n - 1\). Find the minimum possible value of \[ \frac{a_1}{1} + \frac{a_2}{1 + a_1} + \frac{a_3}{1 + a_1 + a_2} + \cdots + \frac{a_n}{1 + a_1 + a_2 + \cdots + a_{n-1}}. \]	[]
CMO-2021-3	https://cms.math.ca/wp-content/uploads/2021/04/CMO-2021-questions-en-4.pdf	At a dinner party there are \(N\) hosts and \(N\) guests, seated around a circular table, where \(N \geq 4\). A pair of two guests will chat with one another if either there is at most one person seated between them or if there are exactly two people between them, at least one of whom is a host. Prove that no matter how the \(2N\) people are seated at the dinner party, at least \(N\) pairs of guests will chat with one another.	[]
CMO-2021-4	https://cms.math.ca/wp-content/uploads/2021/04/CMO-2021-questions-en-4.pdf	A function \(f\) from the positive integers to the positive integers is called \emph{Canadian} if it satisfies \[ \gcd\left(f(f(x)), f(x+y)\right) = \gcd(x, y) \] for all pairs of positive integers \(x\) and \(y\). Find all positive integers \(m\) such that \(f(m) = m\) for all Canadian functions \(f\).	[]
CMO-2021-5	https://cms.math.ca/wp-content/uploads/2021/04/CMO-2021-questions-en-4.pdf	Nina and Tadashi play the following game. Initially, a triple \((a, b, c)\) of nonnegative integers with \(a + b + c = 2021\) is written on a blackboard. Nina and Tadashi then take moves in turn, with Nina first. A player making a move chooses a positive integer \(k\) and one of the three entries on the board; then the player increases the chosen entry by \(k\) and decreases the other two entries by \(k\). A player loses if, on their turn, some entry on the board becomes negative. Find the number of initial triples \((a, b, c)\) for which Tadashi has a winning strategy.	[]
CMO-2022-1	https://cms.math.ca/wp-content/uploads/2022/03/2022CMO-exam-en.pdf	Assume that real numbers \( a \) and \( b \) satisfy \[ ab + \sqrt{ab + 1 + \sqrt{a^2 + b \cdot \sqrt{b^2 + a}}} = 0. \] Find, with proof, the value of \[ a \sqrt{b^2 + a} + b \sqrt{a^2 + b}. \]	[]
CMO-2022-2	https://cms.math.ca/wp-content/uploads/2022/03/2022CMO-exam-en.pdf	Let \( d(k) \) denote the number of positive integer divisors of \( k \). For example, \( d(6) = 4 \) since 6 has 4 positive divisors, namely, 1, 2, 3, and 6. Prove that for all positive integers \( n \), \[ d(1) + d(3) + d(5) + \cdots + d(2n - 1) \leq d(2) + d(4) + d(6) + \cdots + d(2n). \]	[]

CMO-2002-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2002.pdf

Prove that for all positive real numbers \( a, b, \) and \( c \), \[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a + b + c, \] and determine when equality occurs.

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CMO-2002-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2002.pdf

Let \( \Gamma \) be a circle with radius \( r \). Let \( A \) and \( B \) be distinct points on \( \Gamma \) such that \( AB < \sqrt{3}r \). Let the circle with centre \( B \) and radius \( AB \) meet \( \Gamma \) again at \( C \). Let \( P \) be the point inside \( \Gamma \) such that triangle \( ABP \) is equilateral. Finally, let the line \( CP \) meet \( \Gamma \) again at \( Q \). Prove that \( PQ = r \).

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CMO-2002-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2002.pdf

Let \( N = \{0, 1, 2, \ldots\} \). Determine all functions \( f : N \to N \) such that \[ xf(y) + yf(x) = (x + y)f(x^2 + y^2) \] for all \( x \) and \( y \) in \( N \).

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CMO-2003-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2003.pdf

Consider a standard twelve-hour clock whose hour and minute hands move continuously. Let \( m \) be an integer, with \( 1 \leq m \leq 720 \). At precisely \( m \) minutes after 12:00, the angle made by the hour hand and minute hand is exactly \( 1^\circ \). Determine all possible values of \( m \).

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CMO-2003-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2003.pdf

Find the last three digits of the number \( 2003^{2002^{2001}} \).

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CMO-2003-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2003.pdf

Find all real positive solutions (if any) to \[ x^3 + y^3 + z^3 = x + y + z, \quad \text{and} \quad x^2 + y^2 + z^2 = xyz. \]

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CMO-2003-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2003.pdf

Prove that when three circles share the same chord \( AB \), every line through \( A \) different from \( AB \) determines the same ratio \( XY : YZ \), where \( X \) is an arbitrary point different from \( B \) on the first circle while \( Y \) and \( Z \) are the points where \( AX \) intersects the other two circles (labelled so that \( Y \) is between \( X \) and \( Z \)).

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CMO-2003-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2003.pdf

Let \( S \) be a set of \( n \) points in the plane such that any two points of \( S \) are at least 1 unit apart. Prove there is a subset \( T \) of \( S \) with at least \( n/7 \) points such that any two points of \( T \) are at least \( \sqrt{3} \) units apart.

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CMO-2004-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2004.pdf

Find all ordered triples \( (x, y, z) \) of real numbers which satisfy the following system of equations: \[ \begin{cases} xy = z - x - y xz = y - x - z yz = x - y - z \end{cases} \]

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CMO-2004-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2004.pdf

How many ways can 8 mutually non-attacking rooks be placed on the \( 9 \times 9 \) chessboard (shown here) so that all 8 rooks are on squares of the same colour? [Two rooks are said to be attacking each other if they are placed in the same row or column of the board.]

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CMO-2004-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2004.pdf

Let \( A, B, C, D \) be four points on a circle (occurring in clockwise order), with \( AB < AD \) and \( BC > CD \). Let the bisector of angle \( BAD \) meet the circle at \( X \) and the bisector of angle \( BCD \) meet the circle at \( Y \). Consider the hexagon formed by these six points on the circle. If four of the six sides of the hexagon have equal length, prove that \( BD \) must be a diameter of the circle.

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CMO-2004-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2004.pdf

Let \( p \) be an odd prime. Prove that \[ \sum_{k=1}^{p-1} k^{2p-1} \equiv \frac{p(p+1)}{2} \pmod{p^2}. \] [Note that \( a \equiv b \pmod{m} \) means that \( a - b \) is divisible by \( m \).]

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CMO-2004-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2004.pdf

Let \( T \) be the set of all positive integer divisors of \( 2004^{100} \). What is the largest possible number of elements that a subset \( S \) of \( T \) can have if no element of \( S \) is an integer multiple of any other element of \( S \)?

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CMO-2005-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2005.pdf

Consider an equilateral triangle of side length \( n \), which is divided into unit triangles, as shown. Let \( f(n) \) be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in our path share a common edge and the path never travels up (from a lower row to a higher row) or revisits a triangle. An example of one such path is illustrated below for \( n = 5 \). Determine the value of \( f(2005) \).

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CMO-2005-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2005.pdf

Let \( (a, b, c) \) be a Pythagorean triple, i.e., a triplet of positive integers with \( a^2 + b^2 = c^2 \). \begin{itemize} \item[(a)] Prove that \( (c/a + c/b)^2 > 8. \) \item[(b)] Prove that there does not exist any integer \( n \) for which we can find a Pythagorean triple \( (a, b, c) \) satisfying \( (c/a + c/b)^2 = n. \) \end{itemize}

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CMO-2005-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2005.pdf

Let \( S \) be a set of \( n \geq 3 \) points in the interior of a circle. \begin{itemize} \item[(a)] Show that there are three distinct points \( a, b, c \in S \) and three distinct points \( A, B, C \) on the circle such that \( a \) is (strictly) closer to \( A \) than any other point in \( S \), \( b \) is closer to \( B \) than any other point in \( S \) and \( c \) is closer to \( C \) than any other point in \( S \). \item[(b)] Show that for no value of \( n \) can four such points in \( S \) (and corresponding points on the circle) be guaranteed. \end{itemize}

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CMO-2005-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2005.pdf

Let \( ABC \) be a triangle with circumradius \( R \), perimeter \( P \) and area \( K \). Determine the maximum value of \( KP/R^3 \).

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CMO-2005-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2005.pdf

Let’s say that an ordered triple of positive integers \( (a, b, c) \) is \( n \)-powerful if \( a \leq b \leq c \), \( \gcd(a, b, c) = 1 \), and \( a^n + b^n + c^n \) is divisible by \( a + b + c \). For example, \( (1, 2, 2) \) is 5-powerful. \begin{itemize} \item[(a)] Determine all ordered triples (if any) which are \( n \)-powerful for all \( n \geq 1 \). \item[(b)] Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful. \end{itemize} [Note that \( \gcd(a, b, c) \) is the greatest common divisor of \( a, b \) and \( c \).]

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CMO-2006-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2006.pdf

Let \( f(n, k) \) be the number of ways of distributing \( k \) candies to \( n \) children so that each child receives at most 2 candies. For example, if \( n = 3 \), then \( f(3, 7) = 0 \), \( f(3, 6) = 1 \) and \( f(3, 4) = 6 \). Determine the value of \[ f(2006, 1) + f(2006, 4) + f(2006, 7) + \cdots + f(2006, 1000) + f(2006, 1003). \]

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CMO-2006-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2006.pdf

Let \( ABC \) be an acute-angled triangle. Inscribe a rectangle \( DEFG \) in this triangle so that \( D \) is on \( AB \), \( E \) is on \( AC \) and both \( F \) and \( G \) are on \( BC \). Describe the locus of (i.e., the curve occupied by) the intersections of the diagonals of all possible rectangles \( DEFG \).

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CMO-2006-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2006.pdf

In a rectangular array of nonnegative real numbers with \( m \) rows and \( n \) columns, each row and each column contains at least one positive element. Moreover, if a row and a column intersect in a positive element, then the sums of their elements are the same. Prove that \( m = n \).

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CMO-2006-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2006.pdf

Consider a round-robin tournament with \( 2n + 1 \) teams, where each team plays each other team exactly once. We say that three teams \( X \), \( Y \) and \( Z \), form a cycle triplet if \( X \) beats \( Y \), \( Y \) beats \( Z \), and \( Z \) beats \( X \). There are no ties. (a) Determine the minimum number of cycle triplets possible. (b) Determine the maximum number of cycle triplets possible.

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CMO-2006-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2006.pdf

The vertices of a right triangle \( ABC \) inscribed in a circle divide the circumference into three arcs. The right angle is at \( A \), so that the opposite arc \( BC \) is a semicircle while arc \( AB \) and arc \( AC \) are supplementary. To each of the three arcs, we draw a tangent such that its point of tangency is the midpoint of that portion of the tangent intercepted by the extended lines \( AB \) and \( AC \). More precisely, the point \( D \) on arc \( BC \) is the midpoint of the segment joining the points \( D' \) and \( D'' \) where the tangent at \( D \) intersects the extended lines \( AB \) and \( AC \). Similarly for \( E \) on arc \( AC \) and \( F \) on arc \( AB \). Prove that triangle \( DEF \) is equilateral.

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CMO-2007-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2007.pdf

What is the maximum number of non-overlapping \(2 \times 1\) dominoes that can be placed on a \(8 \times 9\) checkerboard if six of them are placed as shown? Each domino must be placed horizontally or vertically so as to cover two adjacent squares of the board. (A diagram shows six dominoes already placed on a checkerboard.)

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CMO-2007-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2007.pdf

You are given a pair of triangles for which (a) two sides of one triangle are equal in length to two sides of the second triangle, and (b) the triangles are similar, but not necessarily congruent. Prove that the ratio of the sides that correspond under the similarity is a number between \( \tfrac{1}{2}(\sqrt{5} - 1) \) and \( \tfrac{1}{2}(\sqrt{5} + 1) \).

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CMO-2007-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2007.pdf

Suppose that \( f \) is a real-valued function for which \[ f(xy) + f(y - x) \geq f(y + x) \] for all real numbers \( x \) and \( y \). (a) Give a nonconstant polynomial that satisfies the condition. (b) Prove that \( f(x) \geq 0 \) for all real \( x \).

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CMO-2007-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2007.pdf

For two real numbers \( a, b \), with \( ab eq 1 \), define the \( * \) operation by \[ a * b = \frac{a + b - 2ab}{1 - ab}. \] Start with a list of \( n \geq 2 \) real numbers whose entries \( x \) all satisfy \( 0 < x < 1 \). Select any two numbers \( a \) and \( b \) in the list; remove them and put the number \( a * b \) at the end of the list, thereby reducing its length by one. Repeat this procedure until a single number remains. (a) Prove that this single number is the same regardless of the choice of pair at each stage. (b) Suppose that the condition on the numbers \( x \) in \( S \) is weakened to \( 0 < x \leq 1 \). What happens if \( S \) contains exactly one 1?

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CMO-2007-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2007.pdf

Let the incircle of triangle \( ABC \) touch sides \( BC, CA \) and \( AB \) at \( D, E \) and \( F \), respectively. Let \( \Gamma, \Gamma_1, \Gamma_2 \) and \( \Gamma_3 \) denote the circumcircles of triangle \( ABC, AEF, BDF \) and \( CDE \) respectively. Let \( \Gamma \) and \( \Gamma_1 \) intersect at \( A \) and \( P \), \( \Gamma \) and \( \Gamma_2 \) intersect at \( B \) and \( Q \), and \( \Gamma \) and \( \Gamma_3 \) intersect at \( C \) and \( R \). (a) Prove that the circles \( \Gamma_1, \Gamma_2 \) and \( \Gamma_3 \) intersect in a common point. (b) Show that \( PD, QE \) and \( RF \) are concurrent.

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CMO-2008-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2008.pdf

Let \(ABCD\) be a convex quadrilateral for which \(AB\) is the longest side. Points \(M\) and \(N\) are located on sides \(AB\) and \(BC\) respectively, so that each of the segments \(AN\) and \(CM\) divides the quadrilateral into two parts of equal area. Prove that the segment \(MN\) bisects the diagonal \(BD\).

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CMO-2008-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2008.pdf

Determine all functions \(f\) defined on the set of rational numbers that take rational values for which \[ f(2f(x) + f(y)) = 2x + y , \] for each \(x\) and \(y\).

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CMO-2008-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2008.pdf

Let \(a, b, c\) be positive real numbers for which \(a + b + c = 1\). Prove that \[ \frac{a - bc}{a + bc} + \frac{b - ca}{b + ca} + \frac{c - ab}{c + ab} \leq \frac{3}{2} . \]

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CMO-2008-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2008.pdf

Determine all functions \(f\) defined on the natural numbers that take values among the natural numbers for which \[ (f(n))^p \equiv n \pmod{f(p)} \] for all \(n \in \mathbb{N}\) and all prime numbers \(p\).

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CMO-2008-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2008.pdf

A \emph{self-avoiding rook walk} on a chessboard (a rectangular grid of unit squares) is a path traced by a sequence of moves parallel to an edge of the board from one unit square to another, such that each begins where the previous move ended and such that no move ever crosses a square that has previously been crossed, \emph{i.e.}, the rook’s path is non-self-intersecting. Let \(R(m,n)\) be the number of self-avoiding rook walks on an \(m \times n\) (\(m\) rows, \(n\) columns) chessboard which begin at the lower-left corner and end at the upper-left corner. For example, \(R(m,1) = 1\) for all natural numbers \(m\); \(R(2,2) = 2\); \(R(3,2) = 4\); \(R(3,3) = 11\). Find a formula for \(R(3,n)\) for each natural number \(n\).

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CMO-2009-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2009.pdf

Given an \( m \times n \) grid with squares coloured either black or white, we say that a black square in the grid is \emph{stranded} if there is some square to its left in the same row that is white and there is some square above it in the same column that is white (see Figure 1). [Figure shows a 4 × 5 grid with no stranded black squares.] Find a closed formula for the number of \( 2 \times n \) grids with no stranded black squares.

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CMO-2009-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2009.pdf

Two circles of different radii are cut out of cardboard. Each circle is subdivided into 200 equal sectors. On each circle 100 sectors are painted white and the other 100 are painted black. The smaller circle is then placed on top of the larger circle, so that their centers coincide. Show that one can rotate the small circle so that the sectors on the two circles line up and at least 100 sectors on the small circle lie over sectors of the same color on the big circle.

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CMO-2009-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2009.pdf

Define \[ f(x, y, z) = \frac{(xy + yz + zx)(x + y + z)}{(x + y)(x + z)(y + z)}. \] Determine the set of real numbers \( r \) for which there exists a triplet \( (x, y, z) \) of positive real numbers satisfying \( f(x, y, z) = r \).

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CMO-2009-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2009.pdf

Find all ordered pairs \( (a, b) \) such that \( a \) and \( b \) are integers and \( 3^a + 7^b \) is a perfect square.

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CMO-2009-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2009.pdf

A set of points is marked on the plane, with the property that any three marked points can be covered with a disk of radius 1. Prove that the set of all marked points can be covered with a disk of radius 1.

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CMO-2010-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2010.pdf

For a positive integer \( n \), an \( n \)-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to \( n \) squares in the \( n \)th row, such that all the left-most squares in each row are aligned vertically. For example, the 5-staircase is shown below. (A diagram of a 5-staircase is shown, with rows of 1 to 5 unit squares aligned on the left.) Let \( f(n) \) denote the minimum number of square tiles required to tile the \( n \)-staircase, where the side lengths of the square tiles can be any positive integer. For example, \( f(2) = 3 \) and \( f(4) = 7 \). (a) Find all \( n \) such that \( f(n) = n \). (b) Find all \( n \) such that \( f(n) = n + 1 \).

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CMO-2010-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2010.pdf

Let \( A, B, P \) be three points on a circle. Prove that if \( a \) and \( b \) are the distances from \( P \) to the tangents at \( A \) and \( B \) and \( c \) is the distance from \( P \) to the chord \( AB \), then \( c^2 = ab \).

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CMO-2010-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2010.pdf

Three speed skaters have a friendly “race” on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does 1 lap a minute, the fastest one does 3.14 laps a minute, and the middle one does \( L \) laps a minute for some \( 1 < L < 3.14 \). The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting point.) Find how many different choices for \( L \) are there such that exactly 117 passings occur before the end of the race. (A passing is defined when one skater passes another one. The beginning and the end of the race when all three skaters are together are not counted as passings.)

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CMO-2010-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2010.pdf

Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex \( P \) and change the colour of \( P \) and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a sequence of operations of this type? (A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex A and vertex B, then B is called a neighbour of A.)

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CMO-2010-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2010.pdf

Let \( P(x) \) and \( Q(x) \) be polynomials with integer coefficients. Let \( a_n = n! + n \). Show that if \( P(a_n)/Q(a_n) \) is an integer for every \( n \), then \( P(n)/Q(n) \) is an integer for every integer \( n \) such that \( Q(n) e 0 \).

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CMO-2011-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2011.pdf

Consider 70-digit numbers \( n \), with the property that each of the digits 1, 2, 3, \ldots, 7 appears in the decimal expansion of \( n \) ten times (and 8, 9, and 0 do not appear). Show that no number of this form can divide another number of this form.

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CMO-2011-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2011.pdf

Let \( ABCD \) be a cyclic quadrilateral whose opposite sides are not parallel, \( X \) the intersection of \( AB \) and \( CD \), and \( Y \) the intersection of \( AD \) and \( BC \). Let the angle bisector of \( \angle X D \) intersect \( AD, BC \) at \( E, F \) respectively and let the angle bisector of \( \angle Y B \) intersect \( AB, CD \) at \( G, H \) respectively. Prove that \( EGFH \) is a parallelogram.

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CMO-2011-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2011.pdf

Amy has divided a square up into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates \( x \), the sum of these numbers. If the total area of the white rectangles equals the total area of the red rectangles, what is the smallest possible value of \( x \)?

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CMO-2011-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2011.pdf

Show that there exists a positive integer \( N \) such that for all integers \( a > N \), there exists a contiguous substring of the decimal expansion of \( a \) that is divisible by 2011. (For instance, if \( a = 153204 \), then 15, 532, and 0 are all contiguous substrings of \( a \). Note that 0 is divisible by 2011.)

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CMO-2011-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2011.pdf

Let \( d \) be a positive integer. Show that for every integer \( S \), there exists an integer \( n > 0 \) and a sequence \( \epsilon_1, \epsilon_2, \ldots, \epsilon_n \), where for any \( k \), \( \epsilon_k = 1 \) or \( \epsilon_k = -1 \), such that \[ S = \epsilon_1(1 + d)^2 + \epsilon_2(1 + 2d)^2 + \epsilon_3(1 + 3d)^2 + \cdots + \epsilon_n(1 + nd)^2. \]

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CMO-2012-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2012.pdf

Let \( x, y \) and \( z \) be positive real numbers. Show that \[ x^2 + xy^2 + xyz^2 \geq 4xyz - 4. \]

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CMO-2012-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2012.pdf

For any positive integers \( n \) and \( k \), let \( L(n,k) \) be the least common multiple of the \( k \) consecutive integers \( n, n+1, \ldots, n+k-1 \). Show that for any integer \( b \), there exist integers \( n \) and \( k \) such that \[ L(n,k) > b \cdot L(n+1,k). \]

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CMO-2012-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2012.pdf

Let \( ABCD \) be a convex quadrilateral and let \( P \) be the point of intersection of \( AC \) and \( BD \). Suppose that \[ AC + AD = BC + BD. \] Prove that the internal angle bisectors of \( \angle ACB \), \( \angle ADB \), and \( \angle APB \) meet at a common point.

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CMO-2012-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2012.pdf

A number of robots are placed on the squares of a finite, rectangular grid of squares. A square can hold any number of robots. Every edge of the grid is classified as either passable or impassable. All edges on the boundary of the grid are impassable. You can give any of the commands \textit{up}, \textit{down}, \textit{left}, or \textit{right}. All of the robots then simultaneously try to move in the specified direction. If the edge adjacent to a robot in that direction is passable, the robot moves across the edge and into the next square. Otherwise, the robot remains on its current square. You can then give another command of \textit{up}, \textit{down}, \textit{left}, or \textit{right}, then another, for as long as you want. Suppose that for any individual robot, and any square on the grid, there is a finite sequence of commands that will move that robot to that square. Prove that you can also give a finite sequence of commands such that \textbf{all} of the robots end up on the same square at the same time.

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CMO-2012-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2012.pdf

A bookshelf contains \( n \) volumes, labelled \( 1 \) to \( n \) in some order. The librarian wishes to put them in the correct order as follows. The librarian selects a volume that is too far to the right, say the volume with label \( k \), takes it out, and inserts it so that it is in the \( k \)-th place. For example, if the bookshelf contains the volumes 1, 3, 2, 4 in that order, the librarian could take out volume 2 and place it in the second position. The books will then be in the correct order 1, 2, 3, 4. (a) Show that if this process is repeated, then, however the librarian makes the selections, all the volumes will eventually be in the correct order. (b) What is the largest number of steps that this process can take?

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CMO-2013-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2013.pdf

Determine all polynomials \( P(x) \) with real coefficients such that \[ (x+1)P(x-1) - (x-1)P(x) \] is a constant polynomial.

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CMO-2013-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2013.pdf

The sequence \( a_1, a_2, \ldots, a_n \) consists of the numbers \( 1, 2, \ldots, n \) in some order. For which positive integers \( n \) is it possible that the \( n+1 \) numbers \( 0, a_1, a_1+a_2, a_1+a_2+a_3, \ldots, a_1 + a_2 + \cdots + a_n \) all have different remainders when divided by \( n+1 \)?

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CMO-2013-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2013.pdf

Let \( G \) be the centroid of a right-angled triangle \( ABC \) with \( \angle BCA = 90^\circ \). Let \( P \) be the point on ray \( AG \) such that \( \angle CPA = \angle CAB \), and let \( Q \) be the point on ray \( BG \) such that \( \angle CQB = \angle ABC \). Prove that the circumcircles of triangles \( AQG \) and \( BPG \) meet at a point on side \( AB \).

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CMO-2013-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2013.pdf

Let \( n \) be a positive integer. For any positive integer \( j \) and positive real number \( r \), define \( f_j(r) \) and \( g_j(r) \) by \[ f_j(r) = \min(jr, n) + \min\left( \frac{j}{r}, n \right), \quad \text{and} \quad g_j(r) = \min(\lfloor jr \rfloor, n) + \min\left( \left\lceil \frac{j}{r} \right\rceil, n \right), \] where \( \lceil x \rceil \) denotes the smallest integer greater than or equal to \( x \). Prove that \[ \sum_{j=1}^n f_j(r) \leq n^2 + n \leq \sum_{j=1}^n g_j(r) \] for all positive real numbers \( r \).

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CMO-2013-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2013.pdf

Let \( O \) denote the circumcentre of an acute-angled triangle \( ABC \). Let point \( P \) on side \( AB \) be such that \( \angle BOP = \angle ABC \), and let point \( Q \) on side \( AC \) be such that \( \angle COQ = \angle ACB \). Prove that the reflection of \( BC \) in the line \( PQ \) is tangent to the circumcircle of triangle \( APQ \).

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CMO-2014-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2014.pdf

Let \( a_1, a_2, \ldots, a_n \) be positive real numbers whose product is 1. Show that the sum \[ \frac{a_1}{1 + a_1} + \frac{a_2}{(1 + a_1)(1 + a_2)} + \frac{a_3}{(1 + a_1)(1 + a_2)(1 + a_3)} + \cdots + \frac{a_n}{(1 + a_1)(1 + a_2) \cdots (1 + a_n)} \] is greater than or equal to \[ \frac{2^n - 1}{2^n}. \]

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CMO-2014-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2014.pdf

Let \( m \) and \( n \) be odd positive integers. Each square of an \( m \) by \( n \) board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of \( m \) and \( n \).

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CMO-2014-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2014.pdf

Let \( p \) be a fixed odd prime. A \( p \)-tuple \( (a_1, a_2, a_3, \ldots, a_p) \) of integers is said to be good if (i) \( 0 \leq a_i \leq p - 1 \) for all \( i \), and (ii) \( a_1 + a_2 + a_3 + \cdots + a_p \) is not divisible by \( p \), and (iii) \( a_1a_2 + a_2a_3 + a_3a_4 + \cdots + a_pa_1 \) is divisible by \( p \). Determine the number of good \( p \)-tuples.

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CMO-2014-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2014.pdf

The quadrilateral \( ABCD \) is inscribed in a circle. The point \( P \) lies in the interior of \( ABCD \), and \( \angle PAB = \angle PBC = \angle PCD = \angle PDA \). The lines \( AD \) and \( BC \) meet at \( Q \), and the lines \( AB \) and \( CD \) meet at \( R \). Prove that the lines \( PQ \) and \( PR \) form the same angle as the diagonals of \( ABCD \).

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CMO-2014-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2014.pdf

Fix positive integers \( n \) and \( k \geq 2 \). A list of \( n \) integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add 1 to all of them or subtract 1 from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least \( n - k + 2 \) of the numbers on the blackboard are all simultaneously divisible by \( k \).

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CMO-2015-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2015.pdf

Let \( \mathbb{N} = \{1, 2, 3, \ldots\} \) be the set of positive integers. Find all functions \( f \), defined on \( \mathbb{N} \) and taking values in \( \mathbb{N} \), such that \[ (n - 1)^2 < f(n)f(f(n)) < n^2 + n \] for every positive integer \( n \).

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CMO-2015-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2015.pdf

Let \( ABC \) be an acute-angled triangle with altitudes \( AD, BE, \) and \( CF \). Let \( H \) be the orthocentre, that is, the point where the altitudes meet. Prove that \[ rac{AB \cdot AC + BC \cdot BA + CA \cdot CB}{AH \cdot AD + BH \cdot BE + CH \cdot CF} \leq 2. \]

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CMO-2015-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2015.pdf

On a \( (4n + 2) imes (4n + 2) \) square grid, a turtle can move between squares sharing a side. The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of \( n \), what is the largest positive integer \( k \) such that there must be a row or column that the turtle has entered at least \( k \) distinct times?

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CMO-2015-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2015.pdf

Let \( ABC \) be an acute-angled triangle with circumcenter \( O \). Let \( \Gamma \) be a circle with centre on the altitude from \( A \) in \( ABC \), passing through vertex \( A \) and points \( P \) and \( Q \) on sides \( AB \) and \( AC \). Assume that \( BP \cdot CQ = AP \cdot AQ \). Prove that \( \Gamma \) is tangent to the circumcircle of triangle \( BOC \).

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CMO-2015-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2015.pdf

Let \( p \) be a prime number for which \( rac{p - 1}{2} \) is also prime, and let \( a, b, c \) be integers not divisible by \( p \). Prove that there are at most \( 1 + \sqrt{2p} \) positive integers \( n \) such that \( n < p \) and \( p \) divides \( a^n + b^n + c^n \).

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CMO-2016-1

https://cms.math.ca/wp-content/uploads/2019/07/exam2016.pdf

The integers \(1, 2, 3, \ldots, 2016\) are written on a board. You can choose any two numbers on the board and replace them with their average. For example, you can replace 1 and 2 with 1.5, or you can replace 1 and 3 with a second copy of 2. After 2015 replacements of this kind, the board will have only one number left on it. (a) Prove that there is a sequence of replacements that will make the final number equal to 2. (b) Prove that there is a sequence of replacements that will make the final number equal to 1000.

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CMO-2016-2

https://cms.math.ca/wp-content/uploads/2019/07/exam2016.pdf

Consider the following system of 10 equations in 10 real variables \(v_1, \ldots, v_{10}\): \[ v_i = 1 + \frac{6v_i^2}{v_1^2 + v_2^2 + \cdots + v_{10}^2} \quad (i = 1, \ldots, 10). \] Find all 10-tuples \((v_1, v_2, \ldots, v_{10})\) that are solutions of this system.

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CMO-2016-3

https://cms.math.ca/wp-content/uploads/2019/07/exam2016.pdf

Find all polynomials \(P(x)\) with integer coefficients such that \(P(P(n) + n)\) is a prime number for infinitely many integers \(n\).

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CMO-2016-4

https://cms.math.ca/wp-content/uploads/2019/07/exam2016.pdf

Let \(A, B,\) and \(F\) be positive integers, and assume \(A < B < 2A\). A flea is at the number 0 on the number line. The flea can move by jumping to the right by \(A\) or by \(B\). Before the flea starts jumping, Lavaman chooses finitely many intervals \(\{m+1, m+2, \ldots, m+A\}\) consisting of \(A\) consecutive positive integers, and places lava at all of the integers in the intervals. The intervals must be chosen so that: (i) any two distinct intervals are disjoint and not adjacent; (ii) there are at least \(F\) positive integers with no lava between any two intervals; and (iii) no lava is placed at any integer less than \(F\). Prove that the smallest \(F\) for which the flea can jump over all the intervals and avoid all the lava, regardless of what Lavaman does, is \[ F = (n - 1)A + B, \] where \(n\) is the positive integer such that \[ \frac{A}{n+1} \leq B - A < \frac{A}{n}. \]

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CMO-2016-5

https://cms.math.ca/wp-content/uploads/2019/07/exam2016.pdf

Let \(\triangle ABC\) be an acute-angled triangle with altitudes \(AD\) and \(BE\) meeting at \(H\). Let \(M\) be the midpoint of segment \(AB\), and suppose that the circumcircles of \(\triangle DEM\) and \(\triangle ABH\) meet at points \(P\) and \(Q\) with \(P\) on the same side of \(CH\) as \(A\). Prove that the lines \(ED\), \(PH\), and \(MQ\) all pass through a single point on the circumcircle of \(\triangle ABC\).

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CMO-2017-1

https://cms.math.ca/wp-content/uploads/2024/03/exam2017.pdf

Let \( a \), \( b \), and \( c \) be non-negative real numbers, no two of which are equal. Prove that \[ \frac{a^2}{(b - c)^2} + \frac{b^2}{(c - a)^2} + \frac{c^2}{(a - b)^2} > 2. \]

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CMO-2017-2

https://cms.math.ca/wp-content/uploads/2024/03/exam2017.pdf

Let \( f \) be a function from the set of positive integers to itself such that, for every \( n \), the number of positive integer divisors of \( n \) is equal to \( f(f(n)) \). For example, \( f(f(6)) = 4 \) and \( f(f(25)) = 3 \). Prove that if \( p \) is prime then \( f(p) \) is also prime.

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CMO-2017-3

https://cms.math.ca/wp-content/uploads/2024/03/exam2017.pdf

Let \( n \) be a positive integer, and define \( S_n = \{1, 2, \ldots, n\} \). Consider a non-empty subset \( T \) of \( S_n \). We say that \( T \) is balanced if the median of \( T \) is equal to the average of \( T \). For example, for \( n = 9 \), each of the subsets \( \{7\}, \{2, 5\}, \{2, 3, 4\}, \{5, 6, 8, 9\}, \{1, 4, 5, 7, 8\} \) is balanced; however, the subsets \( \{2, 4, 5\} \) and \( \{1, 2, 3, 5\} \) are not balanced. For each \( n \geq 1 \), prove that the number of balanced subsets of \( S_n \) is odd. (To define the median of a set of \( k \) numbers, first put the numbers in increasing order; then the median is the middle number if \( k \) is odd, and the average of the two middle numbers if \( k \) is even. For example, the median of \( \{1, 3, 4, 8, 9\} \) is 4, and the median of \( \{1, 3, 4, 7, 8, 9\} \) is \( (4 + 7)/2 = 5.5 \).)

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CMO-2017-4

https://cms.math.ca/wp-content/uploads/2024/03/exam2017.pdf

Points \( P \) and \( Q \) lie inside parallelogram \( ABCD \) and are such that triangles \( ABP \) and \( BCQ \) are equilateral. Prove that the line through \( P \) perpendicular to \( DP \) and the line through \( Q \) perpendicular to \( DQ \) meet on the altitude from \( B \) in triangle \( ABC \).

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CMO-2017-5

https://cms.math.ca/wp-content/uploads/2024/03/exam2017.pdf

One hundred circles of radius one are positioned in the plane so that the area of any triangle formed by the centres of three of these circles is at most 2017. Prove that there is a line intersecting at least three of these circles.

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CMO-2018-1

https://cms.math.ca/wp-content/uploads/2024/03/exam2018.pdf

Consider an arrangement of tokens in the plane, not necessarily at distinct points. We are allowed to apply a sequence of moves of the following kind: Select a pair of tokens at points \( A \) and \( B \) and move both of them to the midpoint of \( A \) and \( B \). We say that an arrangement of \( n \) tokens is \emph{collapsible} if it is possible to end up with all \( n \) tokens at the same point after a finite number of moves. Prove that every arrangement of \( n \) tokens is collapsible if and only if \( n \) is a power of 2.

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CMO-2018-2

https://cms.math.ca/wp-content/uploads/2024/03/exam2018.pdf

Let five points on a circle be labelled \( A, B, C, D, \) and \( E \) in clockwise order. Assume \( AE = DE \) and let \( P \) be the intersection of \( AC \) and \( BD \). Let \( Q \) be the point on the line through \( A \) and \( B \) such that \( A \) is between \( B \) and \( Q \) and \( AQ = DP \). Similarly, let \( R \) be the point on the line through \( C \) and \( D \) such that \( D \) is between \( C \) and \( R \) and \( DR = AP \). Prove that \( PE \) is perpendicular to \( QR \).

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CMO-2018-3

https://cms.math.ca/wp-content/uploads/2024/03/exam2018.pdf

Two positive integers \( a \) and \( b \) are \emph{prime-related} if \( a = pb \) or \( b = pa \) for some prime \( p \). Find all positive integers \( n \), such that \( n \) has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related. Note that 1 and \( n \) are included as divisors.

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CMO-2018-4

https://cms.math.ca/wp-content/uploads/2024/03/exam2018.pdf

Find all polynomials \( p(x) \) with real coefficients that have the following property: There exists a polynomial \( q(x) \) with real coefficients such that \[ p(1) + p(2) + p(3) + \cdots + p(n) = p(n)q(n) \] for all positive integers \( n \).

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CMO-2018-5

https://cms.math.ca/wp-content/uploads/2024/03/exam2018.pdf

Let \( k \) be a given even positive integer. Sarah first picks a positive integer \( N \) greater than 1 and proceeds to alter it as follows: every minute, she chooses a prime divisor \( p \) of the current value of \( N \), and multiplies the current \( N \) by \( p^k - p^{-1} \) to produce the next value of \( N \). Prove that there are infinitely many even positive integers \( k \) such that, no matter what choices Sarah makes, her number \( N \) will at some point be divisible by 2018.

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CMO-2019-1

https://cms.math.ca/wp-content/uploads/2024/03/exam2019.pdf

Amy has drawn three points in a plane, \( A, B, \) and \( C \), such that \( AB = BC = CA = 6 \). Amy is allowed to draw a new point if it is the circumcenter of a triangle whose vertices she has already drawn. For example, she can draw the circumcenter \( O \) of triangle \( ABC \), and then afterwards she can draw the circumcenter of triangle \( ABO \). (a) Prove that Amy can eventually draw a point whose distance from a previously drawn point is greater than 7. (b) Prove that Amy can eventually draw a point whose distance from a previously drawn point is greater than 2019. (Recall that the circumcenter of a triangle is the center of the circle that passes through its three vertices.)

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CMO-2019-2

https://cms.math.ca/wp-content/uploads/2024/03/exam2019.pdf

Let \( a \) and \( b \) be positive integers such that \( a + b^3 \) is divisible by \( a^2 + 3ab + 3b^2 - 1 \). Prove that \( a^2 + 3ab + 3b^2 - 1 \) is divisible by the cube of an integer greater than 1.

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CMO-2019-3

https://cms.math.ca/wp-content/uploads/2024/03/exam2019.pdf

Let \( m \) and \( n \) be positive integers. A \( 2m \times 2n \) grid of squares is coloured in the usual chessboard fashion. Find the number of ways of placing \( mn \) counters on the white squares, at most one counter per square, so that no two counters are on white squares that are diagonally adjacent. An example of a way to place the counters when \( m = 2 \) and \( n = 3 \) is shown below. [Figure shows a 4x6 grid with black and white squares in chessboard pattern, and 6 counters placed on white squares such that no two are diagonally adjacent.]

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CMO-2019-4

https://cms.math.ca/wp-content/uploads/2024/03/exam2019.pdf

Let \( n \) be an integer greater than 1, and let \( a_0, a_1, \ldots, a_n \) be real numbers with \( a_1 = a_{n-1} = 0 \). Prove that for any real number \( k \), \[ |a_0| - |a_n| \leq \sum_{i=0}^{n-2} |a_i - k a_{i+1} - a_{i+2}|. \]

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CMO-2019-5

https://cms.math.ca/wp-content/uploads/2024/03/exam2019.pdf

David and Jacob are playing a game of connecting \( n \geq 3 \) points drawn in a plane. No three of the points are collinear. On each player's turn, he chooses two points to connect by a new line segment. The first player to complete a cycle consisting of an odd number of line segments loses the game. (Both endpoints of each line segment in the cycle must be among the \( n \) given points, not points which arise later as intersections of segments.) Assuming David goes first, determine all \( n \) for which he has a winning strategy.

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CMO-2020-1

https://cms.math.ca/wp-content/uploads/2024/03/exam2020.pdf

Let \( S \) be a set of \( n \geq 3 \) positive real numbers. Show that the largest possible number of distinct integer powers of three that can be written as the sum of three distinct elements of \( S \) is \( n - 2 \).

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CMO-2020-2

https://cms.math.ca/wp-content/uploads/2024/03/exam2020.pdf

A circle is inscribed in a rhombus \( ABCD \). Points \( P \) and \( Q \) vary on line segments \( AB \) and \( AD \), respectively, so that \( PQ \) is tangent to the circle. Show that for all such line segments \( PQ \), the area of triangle \( CPQ \) is constant. (Figure shows rhombus \(ABCD\) with an inscribed circle. Points \(P\) on \(AB\) and \(Q\) on \(AD\) are such that line \(PQ\) is tangent to the circle. Triangle \(CPQ\) is formed.)

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CMO-2020-3

https://cms.math.ca/wp-content/uploads/2024/03/exam2020.pdf

A purse contains a finite number of coins, each with distinct positive integer values. Is it possible that there are exactly 2020 ways to use coins from the purse to make the value 2020?

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CMO-2020-4

https://cms.math.ca/wp-content/uploads/2024/03/exam2020.pdf

Let \( S = \{1, 4, 8, 9, 16, \ldots\} \) be the set of perfect powers of integers, i.e. numbers of the form \( n^k \) where \( n, k \) are positive integers and \( k \geq 2 \). Write \( S = \{a_1, a_2, a_3, \ldots\} \) with terms in increasing order, so that \( a_1 < a_2 < a_3 < \cdots \). Prove that there exist infinitely many integers \( m \) such that 9999 divides the difference \( a_{m+1} - a_m \).

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CMO-2020-5

https://cms.math.ca/wp-content/uploads/2024/03/exam2020.pdf

There are 19,998 people on a social media platform, where any pair of them may or may not be \emph{friends}. For any group of 9,999 people, there are at least 9,999 pairs of them that are friends. What is the least number of friendships, that is, the least number of pairs of people that are friends, that must be among the 19,998 people?

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CMO-2021-1