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... | [] |
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... | [] |
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 t... | [] |
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 low... | [] |
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 +... | [] |
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 close... | [] |
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 a... | [] |
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)... | [] |
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 cy... | [] |
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 ... | [] |
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 checker... | [] |
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)... | [] |
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 ... | [] |
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 ... | [] |
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... | [] |
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 s... | [] |
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 c... | [] |
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 sho... | [] |
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 minu... | [] |
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... | [] |
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... | [] |
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... | [] |
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 + ... | [] |
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}, \texti... | [] |
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 examp... | [] |
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 \( AQG \) a... | [] |
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 \rig... | [] |
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 circum... | [] |
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. Determi... | [] |
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... | [] |
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 ... | [] |
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 o... | [] |
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 ... | [] |
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 t... | [] |
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 lef... | [] |
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 pos... | [] |
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 li... | [] |
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\},... | [] |
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 \emp... | [] |
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... | [] |
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-re... | [] |
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 \( N \).... | [] |
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 th... | [] |
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... | [] |
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 ... | [] |
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 cir... | [] |
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 infini... | [] |
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 ... | [] |
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... | [] |
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_... | [] |
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 ho... | [] |
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... | [] |
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 \( n \),
\[
d(1) + d(3) + d(5) + \cdots + d(2n - 1) \leq d(2) + d(4) + d(6) + \cdots + d(2n).
\] | [] |
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