Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 Bulgaria National Olympiad
Let $ABCD$ be a quadrilateral inscribed in a circle $k$. $AC$ and $BD$ meet at $E$. The rays $\overrightarrow{CB}, \overrightarrow{DA}$ meet at $F$.

Prove that the line through the incenters of $\triangle ABE\,,\, \triangle ABF$ and the line through the incenters of $\triangle CDE\,,\, \triangle CDF$ meet at a point lying on the circle $k$.



Proposed by N. Beluhov
2021 Canada National Olympiad
Let $ABCD$ be a trapezoid with $AB$ parallel to $CD$, $|AB|>|CD|$, and equal edges $|AD|=|BC|$. Let $I$ be the center of the circle tangent to lines $AB$, $AC$ and $BD$, where $A$ and $I$ are on opposite sides of $BD$. Let $J$ be the center of the circle tangent to lines $CD$, $AC$ and $BD$, where $D$ and $J$ are on opposite sides of $AC$. Prove that $|IC|=|JB|$.
2021 Canada National Olympiad
Let $n\geq 2$ be some fixed positive integer and suppose that $a_1, a_2,\dots,a_n$ are positive real numbers satisfying $a_1+a_2+\cdots+a_n=2^n-1$.



Find the minimum possible value of $$\frac{a_1}{1}+\frac{a_2}{1+a_1}+\frac{a_3}{1+a_1+a_2}+\cdots+\frac{a_n}{1+a_1+a_2+\cdots+a_{n-1}}$$
2021 Canada National Olympiad
At a dinner party there are $N$ hosts and $N$ guests, seated around a circular table, where $N\geq 4$. A pair of two guests will chat with one another if either there is at most one person seated between them or if there are exactly two people between them, at least one of whom is a host. Prove that no matter how the $2N$ people are seated at the dinner party, at least $N$ pairs of guests will chat with one another.
2021 Canada National Olympiad
A function $f$ from the positive integers to the positive integers is called 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 $f$.
2021 Canada National Olympiad
Nina and Tadashi play the following game. Initially, a triple $(a, b, c)$ of nonnegative integers with $a+b+c=2021$ is written on a blackboard. Nina and Tadashi then take moves in turn, with Nina first. A player making a move chooses a positive integer $k$ and one of the three entries on the board; then the player increases the chosen entry by $k$ and decreases the other two entries by $k$. A player loses if, on their turn, some entry on the board becomes negative.



Find the number of initial triples $(a, b, c)$ for which Tadashi has a winning strategy.
2020 Canada National Olympiad
There are $n \ge 3$ distinct positive real numbers. Show that there are at most $n-2$ different integer power of three that can be written as the sum of three distinct elements from these $n$ numbers.
2020 Canada National Olympiad
$ABCD$ is a fixed rhombus. Segment $PQ$ is tangent to the inscribed circle of $ABCD$, where $P$ is on side $AB$, $Q$ is on side $AD$. Show that, when segment $PQ$ is moving, the area of $\Delta CPQ$ is a constant.
2020 Canada National Olympiad
There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exactly $2020$ different ways to select the coins in his purse and the sum of these selected coins is $2020$?
2020 Canada National Olympiad
$S= \{1,4,8,9,16,...\} $is the set of perfect integer power. ( $S=\{ n^k| n, k \in Z, k \ge 2 \}$. )We arrange the elements in $S$ into an increasing sequence $\{a_i\}$ . Show that there are infinite many $n$, such that $9999|a_{n+1}-a_n$
2020 Canada National Olympiad
Simple graph $G$ has $19998$ vertices. For any subgraph $\bar G$ of $G$ with $9999$ vertices, $\bar G$ has at least $9999$ edges. Find the minimum number of edges in $G$
2019 Canada National Olympiad
Points $A,B,C$ are on a plane such that $AB=BC=CA=6$. At any step, you may choose any three existing points and draw that triangle's circumcentre. Prove that you can draw a point such that its distance from an previously drawn point is:

$(a)$ greater than 7

$(b)$ greater than 2019
2019 Canada National Olympiad
Let $a,b$ 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.
2019 Canada National Olympiad
You have a $2m$ by $2n$ grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place $mn$ counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.
2019 Canada National Olympiad
Prove that for $n>1$ and real numbers $a_0,a_1,\dots, a_n,k$ with $a_1=a_{n-1}=0$,

$$|a_0|-|a_n|\leq \sum_{i=0}^{n-2}|a_i-ka_{i+1}-a_{i+2}|.$$
2019 Canada National Olympiad
A 2-player game is played on $n\geq 3$ points, where no 3 points are collinear. Each move consists of selecting 2 of the points and drawing a new line segment connecting them. The first player to draw a line segment that creates an odd cycle loses. (An odd cycle must have all its vertices among the $n$ points from the start, so the vertices of the cycle cannot be the intersections of the lines drawn.) Find all $n$ such that the player to move first wins.
2018 Canada National Olympiad
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 collapsible if it is possible to end up with all $n$ tokens at the same point after a finite number of moves. Prove that every arrangement of $n$ tokens is collapsible if and only if $n$ is a power of $2$.
2018 Canada National Olympiad
Let five points on a circle be labelled $A, B, C, D$, and $E$ in clockwise order. Assume $AE = DE$ and let $P$ be the intersection of $AC$ and $BD$. Let $Q$ be the point on the line through $A$ and $B$ such that $A$ is between $B$ and $Q$ and $AQ = DP$ Similarly, let $R$ be the point on the line through $C$ and $D$ such that $D$ is between $C$ and $R$ and $DR = AP$. Prove that $PE$ is perpendicular to $QR$.
2018 Canada National Olympiad
Two positive integers $a$ and $b$ are prime-related if $a = pb$ or $b = pa$ for some prime $p$. Find all positive integers $n$, such that $n$ has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related.



Note that $1$ and $n$ are included as divisors.
2018 Canada National Olympiad
Find all polynomials $p(x)$ with real coefficients that have the following property: there exists a polynomial $q(x)$ with real coefficients such that $$p(1) + p(2) + p(3) +\dots + p(n) = p(n)q(n)$$for all positive integers $n$.
2018 Canada National Olympiad
Let $k$ be a given even positive integer. Sarah first picks a positive integer $N$ greater than $1$ and proceeds to alter it as follows: every minute, she chooses a prime divisor $p$ of the current value of $N$, and multiplies the current $N$ by $p^k -p^{-1}$ to produce the next value of $N$. Prove that there are infinitely many even positive integers $k$ such that, no matter what choices Sarah makes, her number $N$ will at some point be divisible by $2018$.
2017 Canada National Olympiad
For pairwise distinct nonnegative reals $a,b,c$, prove that

$$\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(b-a)^2}>2$$.
2017 Canada National Olympiad
Define a function $f(n)$ from the positive integers to the positive integers such that $f(f(n))$ is the number of positive integer divisors of $n$. Prove that if $p$ is a prime, then $f(p)$ is prime.
2017 Canada National Olympiad
Define $S_n$ as the set ${1,2,\cdots,n}$. A non-empty subset $T_n$ of $S_n$ is called $balanced$ if the average of the elements of $T_n$ is equal to the median of $T_n$. Prove that, for all $n$, the number of balanced subsets $T_n$ is odd.
2017 Canada National Olympiad
Let $ABCD$ be a parallelogram. Points $P$ and $Q$ lie inside $ABCD$ such that $\bigtriangleup ABP$ and $\bigtriangleup{BCQ}$ are equilateral. Prove that the intersection of the line through $P$ perpendicular to $PD$ and the line through $Q$ perpendicular to $DQ$ lies on the altitude from $B$ in $\bigtriangleup{ABC}$.
2017 Canada National Olympiad
There are $100$ circles of radius one in the plane. A triangle formed by the centres of any three given circles has area at most $2017$. Prove that there is a line intersecting at least three of the circles.
2016 Canada National Olympiad
The integers $1, 2, 3, \ldots, 2016$ are written on a board. You can choose any two numbers on the board and replace them with their average. For example, you can replace $1$ and $2$ with $1.5$, or you can replace $1$ and $3$ with a second copy of $2$. After $2015$ replacements of this kind, the board will have only one number left on it.



(a) Prove that there is a sequence of replacements that will make the final number equal to $2$.



(b) Prove that there is a sequence of replacements that will make the final number equal to $1000$.
2016 Canada National Olympiad
Consider the following system of $10$ equations in $10$ real variables $v_1, \ldots, v_{10}$:

$$v_i = 1 + \frac{6v_i^2}{v_1^2 + v_2^2 + \cdots + v_{10}^2} \qquad (i = 1, \ldots, 10).$$Find all $10$-tuples $(v_1, v_2, \ldots , v_{10})$ that are solutions of this system.
2016 Canada National Olympiad
Find all polynomials $P(x)$ with integer coefficients such that $P(P(n) + n)$ is a prime number for infinitely many integers $n$.
2016 Canada National Olympiad
Let $A, B$, and $F$ be positive integers, and assume $A < B < 2A$. A flea is at the number $0$ on the number line. The flea can move by jumping to the right by $A$ or by $B$. Before the flea starts jumping, Lavaman chooses finitely many intervals $\{m+1, m+2, \ldots, m+A\}$ consisting of $A$ consecutive positive integers, and places lava at all of the integers in the intervals. The intervals must be chosen so that:



(i) any two distinct intervals are disjoint and not adjacent;

(ii) there are at least $F$ positive integers with no lava between any two intervals; and

(iii) no lava is placed at any integer less than $F$.



Prove that the smallest $F$ for which the flea can jump over all the intervals and avoid all the lava, regardless of what Lavaman does, is $F = (n-1)A + B$, where $n$ is the positive integer such that $\frac{A}{n+1} \le B-A < \frac{A}{n}$.
2016 Canada National Olympiad
Let $\triangle ABC$ be an acute-angled triangle with altitudes $AD$ and $BE$ meeting at $H$. Let $M$ be the midpoint of segment $AB$, and suppose that the circumcircles of $\triangle DEM$ and $\triangle ABH$ meet at points $P$ and $Q$ with $P$ on the same side of $CH$ as $A$. Prove that the lines $ED, PH,$ and $MQ$ all pass through a single point on the circumcircle of $\triangle ABC$.
2015 Canada National Olympiad
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 $n$.
2015 Canada National Olympiad
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 $$\frac{AB\cdot AC+BC\cdot CA+CA\cdot CB}{AH\cdot AD+BH\cdot BE+CH\cdot CF}\leq 2.$$
2015 Canada National Olympiad
On a $(4n + 2)\times (4n + 2)$ square grid, a turtle can move between squares sharing a side.The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of $n$, what is the largest positive integer $k$ such that there must be a row or column that the turtle has entered at least $k$ distinct times?
2015 Canada National Olympiad
Let $ABC$ be an acute-angled triangle with circumcenter $O$. Let $I$ be a circle with center 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 $I$ is tangent to the circumcircle of triangle $BOC$.
2015 Canada National Olympiad
Let $p$ be a prime number for which $\frac{p-1}{2}$ 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 $a^n+b^n+c^n$.
2014 Canada National Olympiad
Let $a_1,a_2,\dots,a_n$ be positive real numbers whose product is $1$. Show that the sum $$\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}$$ is greater than or equal to $\frac{2^n-1}{2^n}$.
2014 Canada National Olympiad
Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.
2014 Canada National Olympiad
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) $0\le a_i\le p-1$ for all $i$, and

(ii) $a_1+a_2+a_3+\cdots+a_p$ is not divisible by $p$, and

(iii) $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$ is divisible by $p$.





Determine the number of good $p$-tuples.
2014 Canada National Olympiad
The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.
2014 Canada National Olympiad
Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.
2013 Canada National Olympiad
Determine all polynomials $P(x)$ with real coefficients such that

$$(x+1)P(x-1)-(x-1)P(x)$$

is a constant polynomial.
2013 Canada National Olympiad
The sequence $a_1, a_2, \dots, a_n$ consists of the numbers $1, 2, \dots, n$ in some order. For which positive integers $n$ is it possible that the $n+1$ numbers $0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n$ all have different remainders when divided by $n + 1$?
2013 Canada National Olympiad
Let $G$ be the centroid of a right-angled triangle $ABC$ with $\angle BCA = 90^\circ$. Let $P$ be the point on ray $AG$ such that $\angle CPA = \angle CAB$, and let $Q$ be the point on ray $BG$ such that $\angle CQB = \angle ABC$. Prove that the circumcircles of triangles $AQG$ and $BPG$ meet at a point on side $AB$.
2013 Canada National Olympiad
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

$$f_j(r) = \min (jr, n) + \min\left(\frac{j}{r}, n\right), \text{ and } g_j(r) = \min (\lceil jr\rceil, n) + \min \left(\left\lceil\frac{j}{r}\right\rceil, n\right),$$

where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Prove that

$$\sum_{j=1}^n f_j(r)\leq n^2+n\leq \sum_{j=1}^n g_j(r)$$

for all positive real numbers $r$.
2013 Canada National Olympiad
Let $O$ denote the circumcentre of an acute-angled triangle $ABC$. Let point $P$ on side $AB$ be such that $\angle BOP = \angle ABC$, and let point $Q$ on side $AC$ be such that $\angle COQ = \angle ACB$. Prove that the reflection of $BC$ in the line $PQ$ is tangent to the circumcircle of triangle $APQ$.
2012 Canada National Olympiad
Let $x,y$ and $z$ be positive real numbers. Show that $x^2+xy^2+xyz^2\ge 4xyz-4$.
2012 Canada National Olympiad
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 $L(n,k)>bL(n+1,k)$.
2012 Canada National Olympiad
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.
2012 Canada National Olympiad
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 each square 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 up, down, left, or right.



All of the robots then simultaneously try to move in the specified direction. If the edge adjacent to a robot in that direction is passable, the robot moves across the edge and into the next square. Otherwise, the robot remains on its current square. You can then give another command of up, down, left, or right, then another, for as long as you want. Suppose that for any individual robot, and any square on the grid, there is a finite sequence of commands that will move that robot to that square. Prove that you can also give a finite sequence of commands such that all of the robots end up on the same square at the same time.
2012 Canada National Olympiad
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 in the $k$-th position. For example, if the bookshelf contains the volumes $1,3,2,4$ in that order, the librarian could take out volume $2$ and place it in the second position. The books will then be in the correct order $1,2,3,4$.



(a) Show that if this process is repeated, then, however the librarian makes the selections, all the volumes will eventually be in the correct order.

(b) What is the largest number of steps that this process can take?
2011 Canada National Olympiad
Consider $70$-digit numbers with the property that each of the digits $1,2,3,...,7$ appear $10$ times in the decimal expansion of $n$ (and $8,9,0$ do not appear). Show that no number of this form can divide another number of this form.
2011 Canada National Olympiad
Let $ABCD$ be a cyclic quadrilateral with opposite sides not parallel. Let $X$ and $Y$ be the intersections of $AB,CD$ and $AD,BC$ respectively. Let the angle bisector of $\angle AXD$ intersect $AD,BC$ at $E,F$ respectively, and let the angle bisectors of $\angle AYB$ intersect $AB,CD$ at $G,H$ respectively. Prove that $EFGH$ is a parallelogram.
2011 Canada National Olympiad
Amy has divided a square into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates $x$, the sum of these numbers. If the total area of white equals the total area of red, determine the minimum of $x$.
2011 Canada National Olympiad
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 $a$, which is divisible by $2011$.

Note. A contiguous substring of an integer $a$ is an integer with a decimal expansion equivalent to a sequence of consecutive digits taken from the decimal expansion of $a$.
2011 Canada National Olympiad
Let $d$ be a positive integer. Show that for every integer $S$, there exists an integer $n>0$ and a sequence of $n$ integers $\epsilon_1, \epsilon_2,..., \epsilon_n$, where $\epsilon_i = \pm 1$ (not necessarily dependent on each other) for all integers $1\le i\le n$, such that $S=\sum_{i=1}^{n}{\epsilon_i(1+id)^2}$.
2010 Canada National Olympiad
For all natural $n$, 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.

Let $f(n)$ denote the minimum number of square tiles requires to tile the $n$-staircase, where the side lengths of the square tiles can be any natural number. e.g. $f(2)=3$ and $f(4)=7$.

(a) Find all $n$ such that $f(n)=n$.

(b) Find all $n$ such that $f(n) = n+1$.
2010 Canada National Olympiad
Let $A,B,P$ be three points on a circle. Prove that if $a,b$ are the distances from $P$ to the tangents at $A,B$ respectively, and $c$ is the distance from $P$ to the chord $AB$, then $c^2 =ab$.
2010 Canada National Olympiad
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 per minute, the fastest one does $3.14$ laps per minute, and the middle one does $L$ laps a minute for some $1 < L < 3.14$. The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting point.) Determine the number of different choices for $L$ such that exactly $117$ passings occur before the end of the race.

Note: A passing is defined as when one skater passes another one. The beginning and the end of the race when all three skaters are together are not counted as passings.
2010 Canada National Olympiad
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?

Note: A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex $A$ and vertex $B,$ then $A$ and $B$ are neighbours of each other.
2010 Canada National Olympiad
Let $P(x)$ and $Q(x)$ be polynomials with integer coefficients. Let $a_n = n! +n$. Show that if $\frac{P(a_n)}{Q(a_n)}$ is an integer for every $n$, then $\frac{P(n)}{Q(n)}$ is an integer for every integer $n$ such that $Q(n)\neq 0$.
2009 Canada National Olympiad
Given an $m\times n$ grid with unit squares coloured either black or white, a black square in the grid is 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.

Find a closed formula for the number of $2\times n$ grids with no stranded black square.

Note that $n$ is any natural number and the formula must be in terms of $n$ with no other variables.
2009 Canada National Olympiad
Two circles of different radii are cut out of cardboard. Each circle is subdivided into $200$ equal sectors. On each circle $100$ sectors are painted white and the other $100$ are painted black. The smaller circle is then placed on top of the larger circle, so that their centers coincide. Show that one can rotate the small circle so that the sectors on the two circles line up and at least $100$ sectors on the small circle lie over sectors of the same color on the big circle.
2009 Canada National Olympiad
Define $f(x,y,z)=\frac{(xy+yz+zx)(x+y+z)}{(x+y)(y+z)(z+x)}$.

Determine the set of real numbers $r$ for which there exists a triplet of positive real numbers satisfying $f(x,y,z)=r$.
2009 Canada National Olympiad
Find all ordered pairs of integers $(a,b)$ such that $3^a + 7^b$ is a perfect square.
2009 Canada National Olympiad
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 $1$.