kylemontgomery/cmo · Datasets at Hugging Face

id string	source string	problem string	solutions list
CMO-1982-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1982.pdf	Let \( R^n \) be the \( n \)-dimensional Euclidean space. Determine the smallest number \( g(n) \) of points of a set in \( R^n \) such that every point in \( R^n \) is at irrational distance from at least one point in that set.	[]
CMO-1982-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1982.pdf	Let \( p \) be a permutation of the set \( S_n = \{1, 2, \ldots, n\} \). An element \( j \in S_n \) is called a fixed point of \( p \) if \( p(j) = j \). Let \( f_n \) be the number of permutations having no fixed points, and \( g_n \) be the number with exactly one fixed point. Show that \( \|f_n - g_n\| = 1 \).	[]
CMO-1982-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1982.pdf	The altitudes of a tetrahedron \( ABCD \) are extended externally to points \( A', B', C' \) and \( D' \) respectively, where \( AA' = k/h_a, BB' = k/h_b, CC' = k/h_c \) and \( DD' = k/h_d \). Here, \( k \) is a constant and \( h_a \) denotes the length of the altitude of \( ABCD \) from vertex \( A \), etc. Prove that the centroid of the tetrahedron \( A'B'C'D' \) coincides with the centroid of \( ABCD \).	[]
CMO-1983-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1983.pdf	Find all positive integers \( w, x, y \) and \( z \) which satisfy \( w! = x! + y! + z! \).	[]
CMO-1983-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1983.pdf	For each real number \( r \) let \( T_r \) be the transformation of the plane that takes the point \( (x, y) \) into the point \( (2^r x, 2^r x^2 + 2^r y) \). Let \( F \) be the family of all such transformations i.e. \( F = \{ T_r : r ext{ a real number} \} \). Find all curves \( y = f(x) \) whose graphs remain unchanged by every transformation in \( F \).	[]
CMO-1983-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1983.pdf	The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?	[]
CMO-1983-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1983.pdf	Prove that for every prime number \( p \), there are infinitely many positive integers \( n \) such that \( p \) divides \( 2^n - n \).	[]
CMO-1983-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1983.pdf	The geometric mean (G.M.) of \( k \) positive numbers \( a_1, a_2, \ldots, a_k \) is defined to be the (positive) \( k \)-th root of their product. For example, the G.M. of 3, 4, 18 is 6. Show that the G.M. of a set \( S \) of \( n \) positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of \( S \).	[]
CMO-1984-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1984.pdf	Prove that the sum of the squares of 1984 consecutive positive integers cannot be the square of an integer.	[]
CMO-1984-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1984.pdf	Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place: Alice: Are you going to cover your keys? Bob: I would like to, but there are only 7 colours and I have 8 keys. Alice: Yes, but you could always distinguish a key by noticing that the red key next to the green key was different from the red key next to the blue key. Bob: You must be careful what you mean by “next to” or “three keys over from” since you can turn the key ring over and the keys are arranged in a circle. Alice: Even so, you don’t need 8 colours. Problem: What is the smallest number of colours needed to distinguish \( n \) keys if all the keys are to be covered.	[]
CMO-1984-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1984.pdf	An integer is \textit{digitally divisible} if (a) none of its digits is zero; (b) it is divisible by the sum of its digits (\textit{e.g.}, 322 is digitally divisible). Show that there are infinitely many digitally divisible integers.	[]
CMO-1984-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1984.pdf	An acute-angled triangle has unit area. Show that there is a point inside the triangle whose distance from each of the vertices is at least \[ \frac{2}{\sqrt{27}}. \]	[]
CMO-1984-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1984.pdf	Given any 7 real numbers, prove that there are two of them, say \( x \) and \( y \), such that \[ 0 \leq \frac{x - y}{1 + xy} \leq \frac{1}{\sqrt{3}}. \]	[]
CMO-1985-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1985.pdf	The lengths of the sides of a triangle are 6, 8 and 10 units. Prove that there is exactly one straight line which simultaneously bisects the area and perimeter of the triangle.	[]
CMO-1985-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1985.pdf	Prove or disprove that there exists an integer which is doubled when the initial digit is transferred to the end.	[]
CMO-1985-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1985.pdf	Let \( P_1 \) and \( P_2 \) be regular polygons of 1985 sides and perimeters \( x \) and \( y \) respectively. Each side of \( P_1 \) is tangent to a given circle of circumference \( c \) and this circle passes through each vertex of \( P_2 \). Prove \( x + y \ge 2c \). (You may assume that \( an heta \ge heta \) for \( 0 \le heta \le rac{\pi}{2} \).)	[]
CMO-1985-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1985.pdf	Prove that \( 2^{n-1} \) divides \( n! \) if and only if \( n = 2^k - 1 \) for some positive integer \( k \).	[]
CMO-1985-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1985.pdf	Let \( 1 < x_1 < 2 \) and, for \( n = 1, 2, \ldots \), define \( x_{n+1} = 1 + x_n - \frac{1}{2} x_n^2 \). Prove that, for \( n \ge 3 \), \( \|x_n - \sqrt{2}\| < 2^{-n} \).	[]
CMO-1986-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1986.pdf	In the diagram line segments \( AB \) and \( CD \) are of length 1 while angles \( ABC \) and \( CBD \) are \( 90^\circ \) and \( 30^\circ \) respectively. Find \( AC \).	[]
CMO-1986-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1986.pdf	A Mathlon is a competition in which there are \( M \) athletic events. Such a competition was held in which only \( A \), \( B \), and \( C \) participated. In each event \( p_1 \) points were awarded for first place, \( p_2 \) for second and \( p_3 \) for third, where \( p_1 > p_2 > p_3 > 0 \) and \( p_1, p_2, p_3 \) are integers. The final score for \( A \) was 22, for \( B \) was 9 and for \( C \) was also 9. \( B \) won the 100 metres. What is the value of \( M \) and who was second in the high jump?	[]
CMO-1986-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1986.pdf	A chord \( ST \) of constant length slides around a semicircle with diameter \( AB \). \( M \) is the mid-point of \( ST \) and \( P \) is the foot of the perpendicular from \( S \) to \( AB \). Prove that angle \( SPM \) is constant for all positions of \( ST \).	[]
CMO-1986-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1986.pdf	For positive integers \( n \) and \( k \), define \( F(n,k) = \sum_{r=1}^{n} r^{2k-1} \). Prove that \( F(n,1) \) divides \( F(n,k) \).	[]
CMO-1986-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1986.pdf	Let \( u_1, u_2, u_3, \ldots \) be a sequence of integers satisfying the recurrence relation \( u_{n+2} = u_{n+1}^2 - u_n \). Suppose \( u_1 = 39 \) and \( u_2 = 45 \). Prove that 1986 divides infinitely many terms of the sequence.	[]
CMO-1987-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1987.pdf	Find all solutions of \( a^2 + b^2 = n! \) for positive integers \( a, b, n \) with \( a \leq b \) and \( n < 14 \).	[]
CMO-1987-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1987.pdf	The number 1987 can be written as a three digit number \( xyz \) in some base \( b \). If \( x + y + z = 1 + 9 + 8 + 7 \), determine all possible values of \( x, y, z, b \).	[]
CMO-1987-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1987.pdf	Suppose \( ABCD \) is a parallelogram and \( E \) is a point between \( B \) and \( C \) on the line \( BC \). If the triangles \( DEC \), \( BED \) and \( BAD \) are isosceles what are the possible values for the angle \( DAB \)?	[]
CMO-1987-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1987.pdf	On a large, flat field \( n \) people are positioned so that for each person the distances to all the other people are different. Each person holds a water pistol and at a given signal fires and hits the person who is closest. When \( n \) is odd show that there is at least one person left dry. Is this always true when \( n \) is even?	[]
CMO-1987-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1987.pdf	For every positive integer \( n \) show that \[ \lfloor \sqrt{n} + \sqrt{n+1} \rfloor = \lfloor \sqrt{4n+1} \rfloor = \lfloor \sqrt{4n+2} \rfloor = \lfloor \sqrt{4n+3} \rfloor \] where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \) (for example \( \lfloor 2.3 \rfloor = 2 \), \( \lfloor \pi \rfloor = 3 \), \( \lfloor 5 \rfloor = 5 \)).	[]
CMO-1988-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1988.pdf	For what values of \( b \) do the equations: \( 1988x^2 + bx + 8891 = 0 \) and \( 8891x^2 + bx + 1988 = 0 \) have a common root?	[]
CMO-1988-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1988.pdf	A house is in the shape of a triangle, perimeter \( P \) metres and area \( A \) square metres. The garden consists of all the land within 5 metres of the house. How much land do the garden and house together occupy?	[]
CMO-1988-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1988.pdf	Suppose that \( S \) is a finite set of at least five points in the plane; some are coloured red, the others are coloured blue. No subset of three or more similarly coloured points is collinear. Show that there is a triangle (i) whose vertices are all the same colour, and (ii) at least one side of the triangle does not contain a point of the opposite colour.	[]
CMO-1988-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1988.pdf	Let \( x_{n+1} = 4x_n - x_{n-1} \), \( x_0 = 0 \), \( x_1 = 1 \), and \( y_{n+1} = 4y_n - y_{n-1} \), \( y_0 = 1 \), \( y_1 = 2 \). Show for all \( n \geq 0 \) that \( y_n^2 = 3x_n^2 + 1 \).	[]
CMO-1988-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1988.pdf	Let \( S = \{a_1, a_2, \ldots, a_r\} \) denote a sequence of integers. For each non-empty subsequence \( A \) of \( S \), we define \( p(A) \) to be the product of all the integers in \( A \). Let \( m(S) \) be the arithmetic average of \( p(A) \) over all non-empty subsets \( A \) of \( S \). If \( m(S) = 13 \) and if \( m(S \cup \{a_{r+1}\}) = 49 \) for some positive integer \( a_{r+1} \), determine the values of \( a_1, a_2, \ldots, a_r \) and \( a_{r+1} \).	[]
CMO-1989-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1989.pdf	The integers \(1, 2, \ldots, n\) are placed in order so that each value is either strictly bigger than all the preceding values or is strictly smaller than all preceding values. In how many ways can this be done?	[]
CMO-1989-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1989.pdf	Let \(ABC\) be a right angled triangle of area 1. Let \(A'B'C'\) be the points obtained by reflecting \(A, B, C\) respectively, in their opposite sides. Find the area of \( \triangle A'B'C' \).	[]
CMO-1989-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1989.pdf	Define \( \{a_n\}_{n=1} \) as follows: \( a_1 = 1989^{1989} \); \( a_n, n > 1 \), is the sum of the digits of \( a_{n-1} \). What is the value of \( a_5 \)?	[]
CMO-1989-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1989.pdf	There are 5 monkeys and 5 ladders and at the top of each ladder there is a banana. A number of ropes connect the ladders, each rope connects two ladders. No two ropes are attached to the same rung of the same ladder. Each monkey starts at the bottom of a different ladder. The monkeys climb up the ladders but each time they encounter a rope they climb along it to the ladder at the other end of the rope and then continue climbing upwards. Show that, no matter how many ropes there are, each monkey gets a banana.	[]
CMO-1989-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1989.pdf	Given the numbers \(1, 2, 2^2, \ldots, 2^{n-1}\). For a specific permutation \( \sigma = X_1, X_2, \ldots, X_n \) of these numbers we define \( S_1(\sigma) = X_1, S_2(\sigma) = X_1 + X_2, S_3(\sigma) = X_1 + X_2 + X_3, \ldots \) and \( Q(\sigma) = S_1(\sigma) S_2(\sigma) \cdots S_n(\sigma) \). Evaluate \( \sum \frac{1}{Q(\sigma)} \) where the sum is taken over all possible permutations.	[]
CMO-1990-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1990.pdf	A competition involving \( n \geq 2 \) players was held over \( k \) days. On each day, the players received scores of \( 1, 2, 3, \ldots, n \) points with no two players receiving the same score. At the end of the \( k \) days, it was found that each player had exactly 26 points in total. Determine all pairs \( (n, k) \) for which this is possible.	[]
CMO-1990-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1990.pdf	A set of \( \tfrac{1}{2}n(n+1) \) distinct numbers is arranged at random in a triangular array: \[ \begin{array}{ccccccc} & & & \times & & & & & \times & \times & \times & & & \vdots & & \vdots & & \vdots & \times & \times & . & \times & . & \times & \times \end{array} \] Let \( M_k \) be the largest number in the \( k \)-th row from the top. Find the probability that \[ M_1 < M_2 < M_3 < \cdots < M_n. \]	[]
CMO-1990-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1990.pdf	Let \( ABCD \) be a convex quadrilateral inscribed in a circle, and let diagonals \( AC \) and \( BD \) meet at \( X \). The perpendiculars from \( X \) meet the sides \( AB, BC, CD, DA \) at \( A', B', C', D' \) respectively. Prove that \[ \|A'B'\| + \|C'D'\| = \|A'D'\| + \|B'C'\|. \] (\( \|A'B'\| \) is the length of line segment \( A'B' \), etc.)	[]
CMO-1990-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1990.pdf	A particle can travel at speeds up to 2 metres per second along the \( x \)-axis, and up to 1 metre per second elsewhere in the plane. Provide a labelled sketch of the region which can be reached within one second by the particle starting at the origin.	[]
CMO-1990-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1990.pdf	Suppose that a function \( f \) defined on the positive integers satisfies \[ f(1) = 1, \quad f(2) = 2, \] \[ f(n+2) = f(n+2 - f(n+1)) + f(n+1 - f(n)) \quad (n \geq 1). \] (a) Show that (i) \( 0 \leq f(n+1) - f(n) \leq 1 \) (ii) if \( f(n) \) is odd, then \( f(n+1) = f(n) + 1 \). (b) Determine, with justification, all values of \( n \) for which \[ f(n) = 2^k + 1. \]	[]
CMO-1991-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1991.pdf	Show that the equation \( x^2 + y^5 = z^3 \) has infinitely many solutions in integers \( x, y, z \) for which \( xyz e 0 \).	[]
CMO-1991-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1991.pdf	Let \( n \) be a fixed positive integer. Find the sum of all positive integers with the following property: In base 2, it has exactly \( 2n \) digits consisting of \( n \) 1's and \( n \) 0's. (The first digit cannot be 0.)	[]
CMO-1991-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1991.pdf	Let \( C \) be a circle and \( P \) a given point in the plane. Each line through \( P \) which intersects \( C \) determines a chord of \( C \). Show that the midpoints of these chords lie on a circle.	[]
CMO-1991-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1991.pdf	Ten distinct numbers from the set \( \{0,1,2,\ldots,13,14\} \) are to be chosen to fill in the ten circles in the diagram. The absolute values of the differences of the two numbers joined by each segment must be different from the values for all other segments. Is it possible to do this? Justify your answer. (The diagram consists of ten circles arranged in a symmetric pattern with connecting segments between them.)	[]
CMO-1991-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1991.pdf	In the figure, the side length of the large equilateral triangle is 3 and \( f(3) \), the number of parallelograms bounded by sides in the grid, is 15. For the general analogous situation, find a formula for \( f(n) \), the number of parallelograms, for a triangle of side length \( n \). (The figure shows a triangular grid composed of smaller equilateral triangles forming a larger triangle of side length 3.)	[]
CMO-1992-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1992.pdf	Prove that the product of the first \( n \) natural numbers is divisible by the sum of the first \( n \) natural numbers if and only if \( n + 1 \) is not an odd prime.	[]
CMO-1992-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1992.pdf	For \( x, y, z \geq 0 \), establish the inequality \[ x(x - z)^2 + y(y - z)^2 \geq (x - z)(y - z)(x + y - z) \] and determine when equality holds.	[]
CMO-1992-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1992.pdf	In the diagram, \(ABCD\) is a square, with \(U\) and \(V\) interior points of the sides \(AB\) and \(CD\) respectively. Determine all the possible ways of selecting \(U\) and \(V\) so as to maximize the area of the quadrilateral \(PUQV\).	[]
CMO-1992-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1992.pdf	Solve the equation \[ x^2 + \frac{x^2}{(x + 1)^2} = 3. \]	[]
CMO-1992-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1992.pdf	A deck of \(2n + 1\) cards consists of a joker and, for each number between 1 and \(n\) inclusive, two cards marked with that number. The \(2n + 1\) cards are placed in a row, with the joker in the middle. For each \(k\) with \(1 \leq k \leq n\), the two cards numbered \(k\) have exactly \(k - 1\) cards between them. Determine all the values of \(n\) not exceeding 10 for which this arrangement is possible. For which values of \(n\) is it impossible?	[]
CMO-1993-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1993.pdf	Determine a triangle for which the three sides and an altitude are four consecutive integers and for which this altitude partitions the triangle into two right triangles with integer sides. Show that there is only one such triangle.	[]
CMO-1993-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1993.pdf	Show that the number \( x \) is rational if and only if three distinct terms that form a geometric progression can be chosen from the sequence \[ x, x+1, x+2, x+3,\ldots \]	[]
CMO-1993-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1993.pdf	In triangle \( ABC \), the medians to the sides \( AB \) and \( AC \) are perpendicular. Prove that \( \cot B + \cot C \geq \frac{3}{2} \).	[]
CMO-1993-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1993.pdf	A number of schools took part in a tennis tournament. No two players from the same school played against each other. Every two players from different schools played exactly one match against each other. A match between two boys or between two girls was called a single and that between a boy and a girl was called a mixed single. The total number of boys differed from the total number of girls by at most 1. The total number of singles differed from the total number of mixed singles by at most 1. At most how many schools were represented by an odd number of players?	[]
CMO-1993-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1993.pdf	Let \( y_1, y_2, y_3, \ldots \) be a sequence such that \( y_1 = 1 \) and, for \( k > 0 \), is defined by the relationship: \[ y_{2k} = \begin{cases} 2y_k & \text{if } k \text{ is even} 2y_k + 1 & \text{if } k \text{ is odd} \end{cases} \] \[ y_{2k+1} = \begin{cases} 2y_k & \text{if } k \text{ is odd} 2y_k + 1 & \text{if } k \text{ is even} \end{cases} \] Show that the sequence \( y_1, y_2, y_3, \ldots \) takes on every positive integer value exactly once.	[]
CMO-1994-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1994.pdf	Evaluate the sum \[ \sum_{n=1}^{1994} (-1)^n rac{n^2 + n + 1}{n!}. \]	[]
CMO-1994-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1994.pdf	Show that every positive integral power of \( \sqrt{2} - 1 \) is of the form \( \sqrt{m} - \sqrt{m - 1} \) for some positive integer \( m \). (e.g. \( (\sqrt{2} - 1)^2 = 3 - 2\sqrt{2} = \sqrt{9} - \sqrt{8} \)).	[]
CMO-1994-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1994.pdf	Twenty-five men sit around a circular table. Every hour there is a vote, and each must respond yes or no. Each man behaves as follows: on the \( n^{ ext{th}} \) vote, if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the \( (n+1)^{ ext{th}} \) vote as on the \( n^{ ext{th}} \) vote; but if his response is different from that of both his neighbours on the \( n^{ ext{th}} \) vote, then his response on the \( (n+1)^{ ext{th}} \) vote will be different from his response on the \( n^{ ext{th}} \) vote. Prove that, however everybody responded on the first vote, there will be a time after which nobody's response will ever change.	[]
CMO-1994-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1994.pdf	Let \( AB \) be a diameter of a circle \( \Omega \) and \( P \) be any point not on the line through \( A \) and \( B \). Suppose the line through \( P \) and \( A \) cuts \( \Omega \) again in \( U \), and the line through \( P \) and \( B \) cuts \( \Omega \) again in \( V \). (Note that in case of tangency \( U \) may coincide with \( A \) or \( V \) may coincide with \( B \). Also, if \( P \) is on \( \Omega \) then \( P = U = V \).) Suppose that \( \|PU\| = s\|PA\| \) and \( \|PV\| = t\|PB\| \) for some nonnegative real numbers \( s \) and \( t \). Determine the cosine of the angle \( APB \) in terms of \( s \) and \( t \).	[]
CMO-1994-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1994.pdf	Let \( ABC \) be an acute angled triangle. Let \( AD \) be the altitude on \( BC \), and let \( H \) be any interior point on \( AD \). Lines \( BH \) and \( CH \), when extended, intersect \( AC \) and \( AB \) at \( E \) and \( F \), respectively. Prove that \( \angle EDH = \angle FDH \).	[]
CMO-1995-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1995.pdf	Let \( f(x) = \frac{x^4}{x^4 + 3} \). Evaluate the sum \[ f\left( \frac{1}{1996} \right) + f\left( \frac{2}{1996} \right) + f\left( \frac{3}{1996} \right) + \cdots + f\left( \frac{1995}{1996} \right) \]	[]
CMO-1995-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1995.pdf	Let \( a, b, c \) be positive real numbers. Prove that \[ a^a b^b c^c \geq (abc)^{\frac{a+b+c}{3}}. \]	[]
CMO-1995-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1995.pdf	Define a boomerang as a quadrilateral whose opposite sides do not intersect and one of whose internal angles is greater than 180 degrees. (See figure displayed.) Let \( C \) be a convex polygon having 5 sides. Suppose that the interior region of \( C \) is the union of \( q \) quadrilaterals, none of whose interiors intersect one another. Also suppose that \( b \) of these quadrilaterals are boomerangs. Show that \( q \geq b + \frac{q - 2}{4} \).	[]
CMO-1995-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1995.pdf	Let \( n \) be a fixed positive integer. Show that for only nonnegative integers \( k \), the diophantine equation \[ x_1^n + x_2^n + \cdots + x_r^n = y^{nk + 2} \] has infinitely many solutions in positive integers \( x_i \) and \( y \).	[]
CMO-1995-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1995.pdf	Suppose that \( u \) is a real parameter with \( 0 < u < 1 \). Define \[ f(x) = \begin{cases} 0 & \text{if } 0 \leq x \leq u 1 - \left( \sqrt{ux} + \sqrt{(1 - u)(1 - x)} \right)^2 & \text{if } u < x \leq 1 \end{cases} \] and define the sequence \( \{u_n\} \) recursively as follows: \[ u_1 = f(1), \quad \text{and } u_n = f(u_{n-1}) \text{ for all } n > 1. \] Show that there exists a positive integer \( k \) for which \( u_k = 0 \).	[]
CMO-1996-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1996.pdf	If \( \alpha, eta, \gamma \) are the roots of \( x^3 - x - 1 = 0 \), compute \[ rac{1 + \alpha}{1 - \alpha} + rac{1 + eta}{1 - eta} + rac{1 + \gamma}{1 - \gamma}. \]	[]
CMO-1996-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1996.pdf	Find all real solutions to the following system of equations. Carefully justify your answer. \[ egin{cases} \dfrac{4x^2}{1 + 4x^2} = y \dfrac{4y^2}{1 + 4y^2} = z \dfrac{4z^2}{1 + 4z^2} = x \end{cases} \]	[]
CMO-1996-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1996.pdf	We denote an arbitrary permutation of the integers \( 1, \ldots, n \) by \( a_1, \ldots, a_n \). Let \( f(n) \) be the number of these permutations such that (i) \( a_1 = 1 \); (ii) \( \|a_i - a_{i+1}\| \leq 2, \quad i = 1, \ldots, n - 1 \). Determine whether \( f(1996) \) is divisible by 3.	[]
CMO-1996-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1996.pdf	Let \( riangle ABC \) be an isosceles triangle with \( AB = AC \). Suppose that the angle bisector of \( \angle B \) meets \( AC \) at \( D \) and that \( BC = BD + AD \). Determine \( \angle A \).	[]
CMO-1996-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1996.pdf	Let \( r_1, r_2, \ldots, r_m \) be a given set of \( m \) positive rational numbers such that \( \sum_{k=1}^m r_k = 1 \). Define the function \( f \) by \( f(n) = n - \sum_{k=1}^m \lfloor r_k n floor \) for each positive integer \( n \). Determine the minimum and maximum values of \( f(n) \). Here \( \lfloor x floor \) denotes the greatest integer less than or equal to \( x \).	[]
CMO-1997-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1997.pdf	How many pairs of positive integers \( x, y \) are there, with \( x \leq y \), and such that \( \gcd(x,y) = 5! \) and \( \mathrm{lcm}(x,y) = 50! \)? \textbf{Note.} \( \gcd(x,y) \) denotes the greatest common divisor of \( x \) and \( y \), \( \mathrm{lcm}(x,y) \) denotes the least common multiple of \( x \) and \( y \), and \( n! = n \times (n-1) \times \cdots \times 2 \times 1 \).	[]
CMO-1997-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1997.pdf	The closed interval \( A = [0, 50] \) is the union of a finite number of closed intervals, each of length 1. Prove that some of the intervals can be removed so that those remaining are mutually disjoint and have total length \( \geq 25 \). \textbf{Note.} For \( a \leq b \), the closed interval \( [a,b] := \{ x \in \mathbb{R} : a \leq x \leq b \} \) has length \( b - a \); disjoint intervals have empty intersection.	[]
CMO-1997-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1997.pdf	Prove that \[ \frac{1}{1999} < \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdots \frac{1997}{1998} \cdot \frac{1}{44}. \]	[]
CMO-1997-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1997.pdf	The point \( O \) is situated inside the parallelogram \( ABCD \) so that \[ \angle AOB + \angle COD = 180^\circ. \] Prove that \( \angle OBC = \angle ODC \).	[]
CMO-1997-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1997.pdf	Write the sum \[ \sum_{k=0}^{n} \frac{(-1)^k \binom{n}{k}}{k^3 + 9k^2 + 26k + 24} \] in the form \( \frac{p(n)}{q(n)} \), where \( p \) and \( q \) are polynomials with integer coefficients.	[]
CMO-1998-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1998.pdf	Determine the number of real solutions \( a \) to the equation \[ \left\lfloor \tfrac{1}{2} a \right\rfloor + \left\lfloor \tfrac{1}{3} a \right\rfloor + \left\lfloor \tfrac{1}{5} a \right\rfloor = a. \] Here, if \( x \) is a real number, then \( \lfloor x \rfloor \) denotes the greatest integer that is less than or equal to \( x \).	[]
CMO-1998-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1998.pdf	Find all real numbers \( x \) such that \[ x = \left( x - \frac{1}{x} \right)^{1/2} + \left( 1 - \frac{1}{x} \right)^{1/2}. \]	[]
CMO-1998-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1998.pdf	Let \( n \) be a natural number such that \( n \geq 2 \). Show that \[ \frac{1}{n+1} \left( 1 + \frac{1}{3} + \cdots + \frac{1}{2n - 1} \right) > \frac{1}{n} \left( \frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2n} \right). \]	[]
CMO-1998-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1998.pdf	Let \( ABC \) be a triangle with \( \angle BAC = 40^\circ \) and \( \angle ABC = 60^\circ \). Let \( D \) and \( E \) be the points lying on the sides \( AC \) and \( AB \), respectively, such that \( \angle CBD = 40^\circ \) and \( \angle BCE = 70^\circ \). Let \( F \) be the point of intersection of the lines \( BD \) and \( CE \). Show that the line \( AF \) is perpendicular to the line \( BC \).	[]
CMO-1998-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1998.pdf	Let \( m \) be a positive integer. Define the sequence \( a_0, a_1, a_2, \ldots \) by \( a_0 = 0 \), \( a_1 = m \), and \( a_{n+1} = m^2 a_n - a_{n-1} \) for \( n = 1, 2, 3, \ldots \). Prove that an ordered pair \( (a, b) \) of non-negative integers, with \( a \leq b \), gives a solution to the equation \[ \frac{a^2 + b^2}{ab + 1} = m^2 \] if and only if \( (a, b) \) is of the form \( (a_n, a_{n+1}) \) for some \( n \geq 0 \).	[]
CMO-1999-1	https://cms.math.ca/wp-content/uploads/2019/07/exam1999.pdf	Find all real solutions to the equation \( 4x^2 - 40\lfloor x floor + 51 = 0 \). Here, if \( x \) is a real number, then \( \lfloor x floor \) denotes the greatest integer that is less than or equal to \( x \).	[]
CMO-1999-2	https://cms.math.ca/wp-content/uploads/2019/07/exam1999.pdf	Let \( ABC \) be an equilateral triangle of altitude 1. A circle with radius 1 and center on the same side of \( AB \) as \( C \) rolls along the segment \( AB \). Prove that the arc of the circle that is inside the triangle always has the same length.	[]
CMO-1999-3	https://cms.math.ca/wp-content/uploads/2019/07/exam1999.pdf	Determine all positive integers \( n \) with the property that \( n = (d(n))^2 \). Here \( d(n) \) denotes the number of positive divisors of \( n \).	[]
CMO-1999-4	https://cms.math.ca/wp-content/uploads/2019/07/exam1999.pdf	Suppose \( a_1, a_2, \ldots, a_8 \) are eight distinct integers from \( \{1, 2, \ldots, 16, 17\} \). Show that there is an integer \( k > 0 \) such that the equation \( a_i - a_j = k \) has at least three different solutions. Also, find a specific set of 7 distinct integers from \( \{1, 2, \ldots, 16, 17\} \) such that the equation \( a_i - a_j = k \) does not have three distinct solutions for any \( k > 0 \).	[]
CMO-1999-5	https://cms.math.ca/wp-content/uploads/2019/07/exam1999.pdf	Let \( x, y, z \) be non-negative real numbers satisfying \( x + y + z = 1 \). Show that \[ x^2 y + y^2 z + z^2 x \leq \frac{4}{27}, \] and find when equality occurs.	[]
CMO-2000-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2000.pdf	At 12:00 noon, Anne, Beth and Carmen begin running laps around a circular track of length three hundred meters, all starting from the same point on the track. Each jogger maintains a constant speed in one of the two possible directions for an indefinite period of time. Show that if Anne’s speed is different from the other two speeds, then at some later time Anne will be at least one hundred meters from each of the other runners. (Here, distance is measured along the shorter of the two arcs separating two runners.)	[]
CMO-2000-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2000.pdf	A permutation of the integers 1901, 1902, \ldots, 2000 is a sequence \( a_1, a_2, \ldots, a_{100} \) in which each of those integers appears exactly once. Given such a permutation, we form the sequence of partial sums \[ s_1 = a_1, \quad s_2 = a_1 + a_2, \quad s_3 = a_1 + a_2 + a_3, \ldots, \quad s_{100} = a_1 + a_2 + \cdots + a_{100}. \] How many of these permutations will have no terms of the sequence \( s_1, \ldots, s_{100} \) divisible by three?	[]
CMO-2000-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2000.pdf	Let \( A = (a_1, a_2, \ldots, a_{2000}) \) be a sequence of integers each lying in the interval \([-1000, 1000]\). Suppose that the entries in \( A \) sum to 1. Show that some nonempty subsequence of \( A \) sums to zero.	[]
CMO-2000-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2000.pdf	Let \( ABCD \) be a convex quadrilateral with \[ \angle CBD \;=\; 2\angle ADB, \angle ABD \;=\; 2\angle CDB ext{and} \quad AB \;=\; CB. \] Prove that \( AD = CD \).	[]
CMO-2000-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2000.pdf	Suppose that the real numbers \( a_1, a_2, \ldots, a_{100} \) satisfy \[ a_1 \ge a_2 \ge \cdots \ge a_{100} \ge 0, a_1 + a_2 \le 100 ext{and} \quad a_3 + a_4 + \cdots + a_{100} \le 100. \] Determine the maximum possible value of \( a_1^2 + a_2^2 + \cdots + a_{100}^2 \), and find all possible sequences \( a_1, a_2, \ldots, a_{100} \) which achieve this maximum.	[]
CMO-2001-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2001.pdf	Randy: “Hi Rachel, that’s an interesting quadratic equation you have written down. What are its roots?” Rachel: “The roots are two positive integers. One of the roots is my age, and the other root is the age of my younger brother, Jimmy.” Randy: “That is very neat! Let me see if I can figure out how old you and Jimmy are. That shouldn’t be too difficult since all of your coefficients are integers. By the way, I notice that the sum of the three coefficients is a prime number.” Rachel: “Interesting. Now figure out how old I am.” Randy: “Instead, I will guess your age and substitute it for \( x \) in your quadratic equation ... darn, that gives me \( -55 \), and not \( 0 \).” Rachel: “Oh, leave me alone!” (a) Prove that Jimmy is two years old. (b) Determine Rachel’s age.	[]
CMO-2001-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2001.pdf	There is a board numbered \( -10 \) to \( 10 \) as shown. Each square is coloured either red or white, and the sum of the numbers on the red squares is \( n \). Maureen starts with a token on the square labeled 0. She then tosses a fair coin ten times. Every time she flips heads, she moves the token one square to the right. Every time she flips tails, she moves the token one square to the left. At the end of the ten flips, the probability that the token finishes on a red square is a rational number of the form \[ \frac{a}{b}. \] Given that \( a + b = 2001 \), determine the largest possible value for \( n \).	[]
CMO-2001-3	https://cms.math.ca/wp-content/uploads/2019/07/exam2001.pdf	Let \( ABC \) be a triangle with \( AC > AB \). Let \( P \) be the intersection point of the perpendicular bisector of \( BC \) and the internal angle bisector of \( \angle A \). Construct points \( X \) on \( AB \) (extended) and \( Y \) on \( AC \) such that \( PX \) is perpendicular to \( AB \) and \( PY \) is perpendicular to \( AC \). Let \( Z \) be the intersection point of \( XY \) and \( BC \). Determine the value of \( BZ/ZC \).	[]
CMO-2001-4	https://cms.math.ca/wp-content/uploads/2019/07/exam2001.pdf	Let \( n \) be a positive integer. Nancy is given a rectangular table in which each entry is a positive integer. She is permitted to make either of the following two moves: (a) select a row and multiply each entry in this row by \( n \). (b) select a column and subtract \( n \) from each entry in this column. Find all possible values of \( n \) for which the following statement is true: \[ \text{Given any rectangular table, it is possible for Nancy to perform a finite sequence of moves to create a table in which each entry is 0.} \]	[]
CMO-2001-5	https://cms.math.ca/wp-content/uploads/2019/07/exam2001.pdf	Let \( P_0, P_1, P_2 \) be three points on the circumference of a circle with radius 1, where \( P_1P_2 = t < 2 \). For each \( i \geq 3 \), define \( P_i \) to be the centre of the circumcircle of \( \triangle P_{i-1}P_{i-2}P_{i-3} \). (a) Prove that the points \( P_1, P_5, P_9, P_{13}, \ldots \) are collinear. (b) Let \( x \) be the distance from \( P_1 \) to \( P_{1001} \), and let \( y \) be the distance from \( P_{1001} \) to \( P_{2001} \). Determine all values of \( t \) for which \[ \sqrt[500]{x/y} \] is an integer.	[]
CMO-2002-1	https://cms.math.ca/wp-content/uploads/2019/07/exam2002.pdf	Let \( S \) be a subset of \( \{1, 2, \ldots, 9\} \), such that the sums formed by adding each unordered pair of distinct numbers from \( S \) are all different. For example, the subset \( \{1, 2, 3, 5\} \) has this property, but \( \{1, 2, 3, 4, 5\} \) does not, since the pairs \( \{1, 4\} \) and \( \{2, 3\} \) have the same sum, namely 5. What is the maximum number of elements that \( S \) can contain?	[]
CMO-2002-2	https://cms.math.ca/wp-content/uploads/2019/07/exam2002.pdf	Call a positive integer \( n \) practical if every positive integer less than or equal to \( n \) can be written as the sum of distinct divisors of \( n \). For example, the divisors of 6 are \( \mathbf{1, 2, 3,} \) and \( \mathbf{6} \). Since \[ 1 = 1, \quad 2 = 2, \quad 3 = 3, \quad 4 = 1 + 3, \quad 5 = 2 + 3, \quad 6 = 6, \] we see that 6 is practical. Prove that the product of two practical numbers is also practical.	[]

CMO-1982-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1982.pdf

Let \( R^n \) be the \( n \)-dimensional Euclidean space. Determine the smallest number \( g(n) \) of points of a set in \( R^n \) such that every point in \( R^n \) is at irrational distance from at least one point in that set.

[]

CMO-1982-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1982.pdf

Let \( p \) be a permutation of the set \( S_n = \{1, 2, \ldots, n\} \). An element \( j \in S_n \) is called a fixed point of \( p \) if \( p(j) = j \). Let \( f_n \) be the number of permutations having no fixed points, and \( g_n \) be the number with exactly one fixed point. Show that \( |f_n - g_n| = 1 \).

[]

CMO-1982-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1982.pdf

The altitudes of a tetrahedron \( ABCD \) are extended externally to points \( A', B', C' \) and \( D' \) respectively, where \( AA' = k/h_a, BB' = k/h_b, CC' = k/h_c \) and \( DD' = k/h_d \). Here, \( k \) is a constant and \( h_a \) denotes the length of the altitude of \( ABCD \) from vertex \( A \), etc. Prove that the centroid of the tetrahedron \( A'B'C'D' \) coincides with the centroid of \( ABCD \).

[]

CMO-1983-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1983.pdf

Find all positive integers \( w, x, y \) and \( z \) which satisfy \( w! = x! + y! + z! \).

[]

CMO-1983-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1983.pdf

For each real number \( r \) let \( T_r \) be the transformation of the plane that takes the point \( (x, y) \) into the point \( (2^r x, 2^r x^2 + 2^r y) \). Let \( F \) be the family of all such transformations i.e. \( F = \{ T_r : r ext{ a real number} \} \). Find all curves \( y = f(x) \) whose graphs remain unchanged by every transformation in \( F \).

[]

CMO-1983-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1983.pdf

The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?

[]

CMO-1983-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1983.pdf

Prove that for every prime number \( p \), there are infinitely many positive integers \( n \) such that \( p \) divides \( 2^n - n \).

[]

CMO-1983-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1983.pdf

The geometric mean (G.M.) of \( k \) positive numbers \( a_1, a_2, \ldots, a_k \) is defined to be the (positive) \( k \)-th root of their product. For example, the G.M. of 3, 4, 18 is 6. Show that the G.M. of a set \( S \) of \( n \) positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of \( S \).

[]

CMO-1984-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1984.pdf

Prove that the sum of the squares of 1984 consecutive positive integers cannot be the square of an integer.

[]

CMO-1984-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1984.pdf

Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place: Alice: Are you going to cover your keys? Bob: I would like to, but there are only 7 colours and I have 8 keys. Alice: Yes, but you could always distinguish a key by noticing that the red key next to the green key was different from the red key next to the blue key. Bob: You must be careful what you mean by “next to” or “three keys over from” since you can turn the key ring over and the keys are arranged in a circle. Alice: Even so, you don’t need 8 colours. Problem: What is the smallest number of colours needed to distinguish \( n \) keys if all the keys are to be covered.

[]

CMO-1984-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1984.pdf

An integer is \textit{digitally divisible} if (a) none of its digits is zero; (b) it is divisible by the sum of its digits (\textit{e.g.}, 322 is digitally divisible). Show that there are infinitely many digitally divisible integers.

[]

CMO-1984-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1984.pdf

An acute-angled triangle has unit area. Show that there is a point inside the triangle whose distance from each of the vertices is at least \[ \frac{2}{\sqrt{27}}. \]

[]

CMO-1984-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1984.pdf

Given any 7 real numbers, prove that there are two of them, say \( x \) and \( y \), such that \[ 0 \leq \frac{x - y}{1 + xy} \leq \frac{1}{\sqrt{3}}. \]

[]

CMO-1985-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1985.pdf

The lengths of the sides of a triangle are 6, 8 and 10 units. Prove that there is exactly one straight line which simultaneously bisects the area and perimeter of the triangle.

[]

CMO-1985-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1985.pdf

Prove or disprove that there exists an integer which is doubled when the initial digit is transferred to the end.

[]

CMO-1985-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1985.pdf

Let \( P_1 \) and \( P_2 \) be regular polygons of 1985 sides and perimeters \( x \) and \( y \) respectively. Each side of \( P_1 \) is tangent to a given circle of circumference \( c \) and this circle passes through each vertex of \( P_2 \). Prove \( x + y \ge 2c \). (You may assume that \( an heta \ge heta \) for \( 0 \le heta \le rac{\pi}{2} \).)

[]

CMO-1985-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1985.pdf

Prove that \( 2^{n-1} \) divides \( n! \) if and only if \( n = 2^k - 1 \) for some positive integer \( k \).

[]

CMO-1985-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1985.pdf

Let \( 1 < x_1 < 2 \) and, for \( n = 1, 2, \ldots \), define \( x_{n+1} = 1 + x_n - \frac{1}{2} x_n^2 \). Prove that, for \( n \ge 3 \), \( |x_n - \sqrt{2}| < 2^{-n} \).

[]

CMO-1986-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1986.pdf

In the diagram line segments \( AB \) and \( CD \) are of length 1 while angles \( ABC \) and \( CBD \) are \( 90^\circ \) and \( 30^\circ \) respectively. Find \( AC \).

[]

CMO-1986-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1986.pdf

A Mathlon is a competition in which there are \( M \) athletic events. Such a competition was held in which only \( A \), \( B \), and \( C \) participated. In each event \( p_1 \) points were awarded for first place, \( p_2 \) for second and \( p_3 \) for third, where \( p_1 > p_2 > p_3 > 0 \) and \( p_1, p_2, p_3 \) are integers. The final score for \( A \) was 22, for \( B \) was 9 and for \( C \) was also 9. \( B \) won the 100 metres. What is the value of \( M \) and who was second in the high jump?

[]

CMO-1986-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1986.pdf

A chord \( ST \) of constant length slides around a semicircle with diameter \( AB \). \( M \) is the mid-point of \( ST \) and \( P \) is the foot of the perpendicular from \( S \) to \( AB \). Prove that angle \( SPM \) is constant for all positions of \( ST \).

[]

CMO-1986-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1986.pdf

For positive integers \( n \) and \( k \), define \( F(n,k) = \sum_{r=1}^{n} r^{2k-1} \). Prove that \( F(n,1) \) divides \( F(n,k) \).

[]

CMO-1986-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1986.pdf

Let \( u_1, u_2, u_3, \ldots \) be a sequence of integers satisfying the recurrence relation \( u_{n+2} = u_{n+1}^2 - u_n \). Suppose \( u_1 = 39 \) and \( u_2 = 45 \). Prove that 1986 divides infinitely many terms of the sequence.

[]

CMO-1987-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1987.pdf

Find all solutions of \( a^2 + b^2 = n! \) for positive integers \( a, b, n \) with \( a \leq b \) and \( n < 14 \).

[]

CMO-1987-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1987.pdf

The number 1987 can be written as a three digit number \( xyz \) in some base \( b \). If \( x + y + z = 1 + 9 + 8 + 7 \), determine all possible values of \( x, y, z, b \).

[]

CMO-1987-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1987.pdf

Suppose \( ABCD \) is a parallelogram and \( E \) is a point between \( B \) and \( C \) on the line \( BC \). If the triangles \( DEC \), \( BED \) and \( BAD \) are isosceles what are the possible values for the angle \( DAB \)?

[]

CMO-1987-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1987.pdf

On a large, flat field \( n \) people are positioned so that for each person the distances to all the other people are different. Each person holds a water pistol and at a given signal fires and hits the person who is closest. When \( n \) is odd show that there is at least one person left dry. Is this always true when \( n \) is even?

[]

CMO-1987-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1987.pdf

For every positive integer \( n \) show that \[ \lfloor \sqrt{n} + \sqrt{n+1} \rfloor = \lfloor \sqrt{4n+1} \rfloor = \lfloor \sqrt{4n+2} \rfloor = \lfloor \sqrt{4n+3} \rfloor \] where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \) (for example \( \lfloor 2.3 \rfloor = 2 \), \( \lfloor \pi \rfloor = 3 \), \( \lfloor 5 \rfloor = 5 \)).

[]

CMO-1988-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1988.pdf

For what values of \( b \) do the equations: \( 1988x^2 + bx + 8891 = 0 \) and \( 8891x^2 + bx + 1988 = 0 \) have a common root?

[]

CMO-1988-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1988.pdf

A house is in the shape of a triangle, perimeter \( P \) metres and area \( A \) square metres. The garden consists of all the land within 5 metres of the house. How much land do the garden and house together occupy?

[]

CMO-1988-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1988.pdf

Suppose that \( S \) is a finite set of at least five points in the plane; some are coloured red, the others are coloured blue. No subset of three or more similarly coloured points is collinear. Show that there is a triangle (i) whose vertices are all the same colour, and (ii) at least one side of the triangle does not contain a point of the opposite colour.

[]

CMO-1988-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1988.pdf

Let \( x_{n+1} = 4x_n - x_{n-1} \), \( x_0 = 0 \), \( x_1 = 1 \), and \( y_{n+1} = 4y_n - y_{n-1} \), \( y_0 = 1 \), \( y_1 = 2 \). Show for all \( n \geq 0 \) that \( y_n^2 = 3x_n^2 + 1 \).

[]

CMO-1988-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1988.pdf

Let \( S = \{a_1, a_2, \ldots, a_r\} \) denote a sequence of integers. For each non-empty subsequence \( A \) of \( S \), we define \( p(A) \) to be the product of all the integers in \( A \). Let \( m(S) \) be the arithmetic average of \( p(A) \) over all non-empty subsets \( A \) of \( S \). If \( m(S) = 13 \) and if \( m(S \cup \{a_{r+1}\}) = 49 \) for some positive integer \( a_{r+1} \), determine the values of \( a_1, a_2, \ldots, a_r \) and \( a_{r+1} \).

[]

CMO-1989-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1989.pdf

The integers \(1, 2, \ldots, n\) are placed in order so that each value is either strictly bigger than all the preceding values or is strictly smaller than all preceding values. In how many ways can this be done?

[]

CMO-1989-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1989.pdf

Let \(ABC\) be a right angled triangle of area 1. Let \(A'B'C'\) be the points obtained by reflecting \(A, B, C\) respectively, in their opposite sides. Find the area of \( \triangle A'B'C' \).

[]

CMO-1989-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1989.pdf

Define \( \{a_n\}_{n=1} \) as follows: \( a_1 = 1989^{1989} \); \( a_n, n > 1 \), is the sum of the digits of \( a_{n-1} \). What is the value of \( a_5 \)?

[]

CMO-1989-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1989.pdf

There are 5 monkeys and 5 ladders and at the top of each ladder there is a banana. A number of ropes connect the ladders, each rope connects two ladders. No two ropes are attached to the same rung of the same ladder. Each monkey starts at the bottom of a different ladder. The monkeys climb up the ladders but each time they encounter a rope they climb along it to the ladder at the other end of the rope and then continue climbing upwards. Show that, no matter how many ropes there are, each monkey gets a banana.

[]

CMO-1989-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1989.pdf

Given the numbers \(1, 2, 2^2, \ldots, 2^{n-1}\). For a specific permutation \( \sigma = X_1, X_2, \ldots, X_n \) of these numbers we define \( S_1(\sigma) = X_1, S_2(\sigma) = X_1 + X_2, S_3(\sigma) = X_1 + X_2 + X_3, \ldots \) and \( Q(\sigma) = S_1(\sigma) S_2(\sigma) \cdots S_n(\sigma) \). Evaluate \( \sum \frac{1}{Q(\sigma)} \) where the sum is taken over all possible permutations.

[]

CMO-1990-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1990.pdf

A competition involving \( n \geq 2 \) players was held over \( k \) days. On each day, the players received scores of \( 1, 2, 3, \ldots, n \) points with no two players receiving the same score. At the end of the \( k \) days, it was found that each player had exactly 26 points in total. Determine all pairs \( (n, k) \) for which this is possible.

[]

CMO-1990-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1990.pdf

A set of \( \tfrac{1}{2}n(n+1) \) distinct numbers is arranged at random in a triangular array: \[ \begin{array}{ccccccc} & & & \times & & & & & \times & \times & \times & & & \vdots & & \vdots & & \vdots & \times & \times & . & \times & . & \times & \times \end{array} \] Let \( M_k \) be the largest number in the \( k \)-th row from the top. Find the probability that \[ M_1 < M_2 < M_3 < \cdots < M_n. \]

[]

CMO-1990-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1990.pdf

Let \( ABCD \) be a convex quadrilateral inscribed in a circle, and let diagonals \( AC \) and \( BD \) meet at \( X \). The perpendiculars from \( X \) meet the sides \( AB, BC, CD, DA \) at \( A', B', C', D' \) respectively. Prove that \[ |A'B'| + |C'D'| = |A'D'| + |B'C'|. \] (\( |A'B'| \) is the length of line segment \( A'B' \), etc.)

[]

CMO-1990-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1990.pdf

A particle can travel at speeds up to 2 metres per second along the \( x \)-axis, and up to 1 metre per second elsewhere in the plane. Provide a labelled sketch of the region which can be reached within one second by the particle starting at the origin.

[]

CMO-1990-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1990.pdf

Suppose that a function \( f \) defined on the positive integers satisfies \[ f(1) = 1, \quad f(2) = 2, \] \[ f(n+2) = f(n+2 - f(n+1)) + f(n+1 - f(n)) \quad (n \geq 1). \] (a) Show that (i) \( 0 \leq f(n+1) - f(n) \leq 1 \) (ii) if \( f(n) \) is odd, then \( f(n+1) = f(n) + 1 \). (b) Determine, with justification, all values of \( n \) for which \[ f(n) = 2^k + 1. \]

[]

CMO-1991-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1991.pdf

Show that the equation \( x^2 + y^5 = z^3 \) has infinitely many solutions in integers \( x, y, z \) for which \( xyz e 0 \).

[]

CMO-1991-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1991.pdf

Let \( n \) be a fixed positive integer. Find the sum of all positive integers with the following property: In base 2, it has exactly \( 2n \) digits consisting of \( n \) 1's and \( n \) 0's. (The first digit cannot be 0.)

[]

CMO-1991-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1991.pdf

Let \( C \) be a circle and \( P \) a given point in the plane. Each line through \( P \) which intersects \( C \) determines a chord of \( C \). Show that the midpoints of these chords lie on a circle.

[]

CMO-1991-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1991.pdf

Ten distinct numbers from the set \( \{0,1,2,\ldots,13,14\} \) are to be chosen to fill in the ten circles in the diagram. The absolute values of the differences of the two numbers joined by each segment must be different from the values for all other segments. Is it possible to do this? Justify your answer. (The diagram consists of ten circles arranged in a symmetric pattern with connecting segments between them.)

[]

CMO-1991-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1991.pdf

In the figure, the side length of the large equilateral triangle is 3 and \( f(3) \), the number of parallelograms bounded by sides in the grid, is 15. For the general analogous situation, find a formula for \( f(n) \), the number of parallelograms, for a triangle of side length \( n \). (The figure shows a triangular grid composed of smaller equilateral triangles forming a larger triangle of side length 3.)

[]

CMO-1992-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1992.pdf

Prove that the product of the first \( n \) natural numbers is divisible by the sum of the first \( n \) natural numbers if and only if \( n + 1 \) is not an odd prime.

[]

CMO-1992-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1992.pdf

For \( x, y, z \geq 0 \), establish the inequality \[ x(x - z)^2 + y(y - z)^2 \geq (x - z)(y - z)(x + y - z) \] and determine when equality holds.

[]

CMO-1992-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1992.pdf

In the diagram, \(ABCD\) is a square, with \(U\) and \(V\) interior points of the sides \(AB\) and \(CD\) respectively. Determine all the possible ways of selecting \(U\) and \(V\) so as to maximize the area of the quadrilateral \(PUQV\).

[]

CMO-1992-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1992.pdf

Solve the equation \[ x^2 + \frac{x^2}{(x + 1)^2} = 3. \]

[]

CMO-1992-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1992.pdf

A deck of \(2n + 1\) cards consists of a joker and, for each number between 1 and \(n\) inclusive, two cards marked with that number. The \(2n + 1\) cards are placed in a row, with the joker in the middle. For each \(k\) with \(1 \leq k \leq n\), the two cards numbered \(k\) have exactly \(k - 1\) cards between them. Determine all the values of \(n\) not exceeding 10 for which this arrangement is possible. For which values of \(n\) is it impossible?

[]

CMO-1993-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1993.pdf

Determine a triangle for which the three sides and an altitude are four consecutive integers and for which this altitude partitions the triangle into two right triangles with integer sides. Show that there is only one such triangle.

[]

CMO-1993-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1993.pdf

Show that the number \( x \) is rational if and only if three distinct terms that form a geometric progression can be chosen from the sequence \[ x, x+1, x+2, x+3,\ldots \]

[]

CMO-1993-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1993.pdf

In triangle \( ABC \), the medians to the sides \( AB \) and \( AC \) are perpendicular. Prove that \( \cot B + \cot C \geq \frac{3}{2} \).

[]

CMO-1993-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1993.pdf

A number of schools took part in a tennis tournament. No two players from the same school played against each other. Every two players from different schools played exactly one match against each other. A match between two boys or between two girls was called a single and that between a boy and a girl was called a mixed single. The total number of boys differed from the total number of girls by at most 1. The total number of singles differed from the total number of mixed singles by at most 1. At most how many schools were represented by an odd number of players?

[]

CMO-1993-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1993.pdf

Let \( y_1, y_2, y_3, \ldots \) be a sequence such that \( y_1 = 1 \) and, for \( k > 0 \), is defined by the relationship: \[ y_{2k} = \begin{cases} 2y_k & \text{if } k \text{ is even} 2y_k + 1 & \text{if } k \text{ is odd} \end{cases} \] \[ y_{2k+1} = \begin{cases} 2y_k & \text{if } k \text{ is odd} 2y_k + 1 & \text{if } k \text{ is even} \end{cases} \] Show that the sequence \( y_1, y_2, y_3, \ldots \) takes on every positive integer value exactly once.

[]

CMO-1994-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1994.pdf

Evaluate the sum \[ \sum_{n=1}^{1994} (-1)^n rac{n^2 + n + 1}{n!}. \]

[]

CMO-1994-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1994.pdf

Show that every positive integral power of \( \sqrt{2} - 1 \) is of the form \( \sqrt{m} - \sqrt{m - 1} \) for some positive integer \( m \). (e.g. \( (\sqrt{2} - 1)^2 = 3 - 2\sqrt{2} = \sqrt{9} - \sqrt{8} \)).

[]

CMO-1994-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1994.pdf

Twenty-five men sit around a circular table. Every hour there is a vote, and each must respond yes or no. Each man behaves as follows: on the \( n^{ ext{th}} \) vote, if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the \( (n+1)^{ ext{th}} \) vote as on the \( n^{ ext{th}} \) vote; but if his response is different from that of both his neighbours on the \( n^{ ext{th}} \) vote, then his response on the \( (n+1)^{ ext{th}} \) vote will be different from his response on the \( n^{ ext{th}} \) vote. Prove that, however everybody responded on the first vote, there will be a time after which nobody's response will ever change.

[]

CMO-1994-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1994.pdf

Let \( AB \) be a diameter of a circle \( \Omega \) and \( P \) be any point not on the line through \( A \) and \( B \). Suppose the line through \( P \) and \( A \) cuts \( \Omega \) again in \( U \), and the line through \( P \) and \( B \) cuts \( \Omega \) again in \( V \). (Note that in case of tangency \( U \) may coincide with \( A \) or \( V \) may coincide with \( B \). Also, if \( P \) is on \( \Omega \) then \( P = U = V \).) Suppose that \( |PU| = s|PA| \) and \( |PV| = t|PB| \) for some nonnegative real numbers \( s \) and \( t \). Determine the cosine of the angle \( APB \) in terms of \( s \) and \( t \).

[]

CMO-1994-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1994.pdf

Let \( ABC \) be an acute angled triangle. Let \( AD \) be the altitude on \( BC \), and let \( H \) be any interior point on \( AD \). Lines \( BH \) and \( CH \), when extended, intersect \( AC \) and \( AB \) at \( E \) and \( F \), respectively. Prove that \( \angle EDH = \angle FDH \).

[]

CMO-1995-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1995.pdf

Let \( f(x) = \frac{x^4}{x^4 + 3} \). Evaluate the sum \[ f\left( \frac{1}{1996} \right) + f\left( \frac{2}{1996} \right) + f\left( \frac{3}{1996} \right) + \cdots + f\left( \frac{1995}{1996} \right) \]

[]

CMO-1995-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1995.pdf

Let \( a, b, c \) be positive real numbers. Prove that \[ a^a b^b c^c \geq (abc)^{\frac{a+b+c}{3}}. \]

[]

CMO-1995-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1995.pdf

Define a boomerang as a quadrilateral whose opposite sides do not intersect and one of whose internal angles is greater than 180 degrees. (See figure displayed.) Let \( C \) be a convex polygon having 5 sides. Suppose that the interior region of \( C \) is the union of \( q \) quadrilaterals, none of whose interiors intersect one another. Also suppose that \( b \) of these quadrilaterals are boomerangs. Show that \( q \geq b + \frac{q - 2}{4} \).

[]

CMO-1995-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1995.pdf

Let \( n \) be a fixed positive integer. Show that for only nonnegative integers \( k \), the diophantine equation \[ x_1^n + x_2^n + \cdots + x_r^n = y^{nk + 2} \] has infinitely many solutions in positive integers \( x_i \) and \( y \).

[]

CMO-1995-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1995.pdf

Suppose that \( u \) is a real parameter with \( 0 < u < 1 \). Define \[ f(x) = \begin{cases} 0 & \text{if } 0 \leq x \leq u 1 - \left( \sqrt{ux} + \sqrt{(1 - u)(1 - x)} \right)^2 & \text{if } u < x \leq 1 \end{cases} \] and define the sequence \( \{u_n\} \) recursively as follows: \[ u_1 = f(1), \quad \text{and } u_n = f(u_{n-1}) \text{ for all } n > 1. \] Show that there exists a positive integer \( k \) for which \( u_k = 0 \).

[]

CMO-1996-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1996.pdf

If \( \alpha, eta, \gamma \) are the roots of \( x^3 - x - 1 = 0 \), compute \[ rac{1 + \alpha}{1 - \alpha} + rac{1 + eta}{1 - eta} + rac{1 + \gamma}{1 - \gamma}. \]

[]

CMO-1996-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1996.pdf

Find all real solutions to the following system of equations. Carefully justify your answer. \[ egin{cases} \dfrac{4x^2}{1 + 4x^2} = y \dfrac{4y^2}{1 + 4y^2} = z \dfrac{4z^2}{1 + 4z^2} = x \end{cases} \]

[]

CMO-1996-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1996.pdf

We denote an arbitrary permutation of the integers \( 1, \ldots, n \) by \( a_1, \ldots, a_n \). Let \( f(n) \) be the number of these permutations such that (i) \( a_1 = 1 \); (ii) \( |a_i - a_{i+1}| \leq 2, \quad i = 1, \ldots, n - 1 \). Determine whether \( f(1996) \) is divisible by 3.

[]

CMO-1996-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1996.pdf

Let \( riangle ABC \) be an isosceles triangle with \( AB = AC \). Suppose that the angle bisector of \( \angle B \) meets \( AC \) at \( D \) and that \( BC = BD + AD \). Determine \( \angle A \).

[]

CMO-1996-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1996.pdf

Let \( r_1, r_2, \ldots, r_m \) be a given set of \( m \) positive rational numbers such that \( \sum_{k=1}^m r_k = 1 \). Define the function \( f \) by \( f(n) = n - \sum_{k=1}^m \lfloor r_k n floor \) for each positive integer \( n \). Determine the minimum and maximum values of \( f(n) \). Here \( \lfloor x floor \) denotes the greatest integer less than or equal to \( x \).

[]

CMO-1997-1

https://cms.math.ca/wp-content/uploads/2019/07/exam1997.pdf

How many pairs of positive integers \( x, y \) are there, with \( x \leq y \), and such that \( \gcd(x,y) = 5! \) and \( \mathrm{lcm}(x,y) = 50! \)? \textbf{Note.} \( \gcd(x,y) \) denotes the greatest common divisor of \( x \) and \( y \), \( \mathrm{lcm}(x,y) \) denotes the least common multiple of \( x \) and \( y \), and \( n! = n \times (n-1) \times \cdots \times 2 \times 1 \).

[]

CMO-1997-2

https://cms.math.ca/wp-content/uploads/2019/07/exam1997.pdf

The closed interval \( A = [0, 50] \) is the union of a finite number of closed intervals, each of length 1. Prove that some of the intervals can be removed so that those remaining are mutually disjoint and have total length \( \geq 25 \). \textbf{Note.} For \( a \leq b \), the closed interval \( [a,b] := \{ x \in \mathbb{R} : a \leq x \leq b \} \) has length \( b - a \); disjoint intervals have empty intersection.

[]

CMO-1997-3

https://cms.math.ca/wp-content/uploads/2019/07/exam1997.pdf

Prove that \[ \frac{1}{1999} < \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdots \frac{1997}{1998} \cdot \frac{1}{44}. \]

[]

CMO-1997-4

https://cms.math.ca/wp-content/uploads/2019/07/exam1997.pdf

The point \( O \) is situated inside the parallelogram \( ABCD \) so that \[ \angle AOB + \angle COD = 180^\circ. \] Prove that \( \angle OBC = \angle ODC \).

[]

CMO-1997-5

https://cms.math.ca/wp-content/uploads/2019/07/exam1997.pdf

Write the sum \[ \sum_{k=0}^{n} \frac{(-1)^k \binom{n}{k}}{k^3 + 9k^2 + 26k + 24} \] in the form \( \frac{p(n)}{q(n)} \), where \( p \) and \( q \) are polynomials with integer coefficients.

[]

CMO-1998-1