Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

In each unit cell of a finite set of cells of an infinite checkered board, an integer is written so that the sum of the numbers in each row, as well as in each column, is divided by $2002$. Prove that every number $\alpha$ can be replaced by a certain number $\alpha'$ , divisible by $2002$ so that $|\alpha-\alpha'| <2002$ and the sum of the numbers in all rows, and in all columns will not change.

On a rectangular board with $m$ rows and $n$ columns, where $m\leq n$, some squares are coloured black in a way that no two rows are alike. Find the biggest integer $k$ such that for every possible colouring to start with one can always color $k$ columns entirely red in such a way that no two rows are still alike.

The centre of a circle is the origin of $N$ vectors whose ends divide the circle in $N$ equal arcs . Some of the vectors are blue and some are red . We calculate the sum of the angles formed between each pair consisting of a red vector and a blue vector (the angle being measured anticlockwise from red to blue) and divide this sum by the total number of such angles . Prove that the "mean angle" thus obtained is $180^o$. (V. P roizvolov)

$S=\{1,2,...,N\}$. $A_1,A_2,A_3,A_4\subseteq S$, each having cardinality $500$. $\forall x,y\in S$, $\exists i\in\{1,2,3,4\}$, $x,y\in A_i$. Determine the maximal value of $N$.

Let $r$ be a positive integer. Show that if a graph $G$ has no cycles of length at most $2r$, then it has at most $|V|^{2016}$ cycles of length exactly $2016r$, where $|V|$ denotes the number of vertices in the graph $G$.

Marine biologists are studying a new species of shellfish whose first generation consists of $100$ shellfish, and their colony reproduces as follows: if a given generation consists of $N$ shellfish (where $5\mid N$ always holds), they divide themselves into $N/5$ groups of $5$ shellfish each. Each group collectively produces $15$ offspring, who form the next generation. Some of the shellfish contain a pearl, but a shellfish can only contain a pearl if none of its direct ancestors contained a pearl. The value of a pearl is determined by the generation of the shellfish containing it: in the $n^{\mathrm{th}}$ generation, its value is $1/3^n$. Find the maximum possible total value of the pearls in the colony. [i]Proposed by: Csongor Beke, Cambridge[/i]

Given positive numbers $a_1, a_2, ..., a_m, b_1, b_2, ..., b_n$. Is known that $$a_1+a_2+...+a_m=b_1+b_2+...+b_n.$$ Prove that you can fill an empty table with $m$ rows and $n$ columns with no more than $(m+n-1)$ positive number in such a way, that for all $i,j$ the sum of the numbers in the $i$-th row will equal to $a_i$, and the sum of the numbers in the $j$-th column -- to $b_j$.

Find the number of positive integers less than $2019$ that are neither multiples of $3$ nor have any digits that are multiples of $3$.

Let $A$ and $B$ be finite sets of real numbers and let $x$ be an element of $A+B$. Prove that \[|A\cap (x-B)|\leq \frac{|A-B|^2}{|A+B|}\] where $A+B=\{a+b: a\in A, b\in B\}$, $x-B=\{x-b: b\in B\}$ and $A-B=\{a-b: a\in A, b\in B\}$.

Initially, all edges of the $K_{2024}$ are painted with $13$ different colors. If there exist $k$ colors such that the subgraph constructed by the edges which are colored with these $k$ colors is connected no matter how the initial coloring was, find the minimum value of $k$.

The circuit diagram drawn (see figure ) contains a battery $B$, a lamp $L$ and five switches $S_1$ to $S_5$. The probability that switch $S_3$ is closed (makes contact) is $\frac23$, for the other four switches that probability is $\frac12$ (the probabilities are mutually independent). Calculate the probability that the light is on. [asy] unitsize (2 cm); draw((-1,1)--(-0.5,1)); draw((-0.25,1)--(1,1)--(1,0.25)); draw((1,-0.25)--(1,-1)--(0.05,-1)); draw((-0.05,-1)--(-1,-1)--(-1,0.25)); draw((-1,0.5)--(-1,1)); draw((-1,1)--(-0.5,0.5)); draw((-0.25,0.25)--(0,0)); draw((-1,0)--(-0.75,0)); draw((-0.5,0)--(0,0)); draw((0,1)--(0,0.75)); draw((0,0.5)--(0,0)); draw((-0.25,1)--(-0.5,1.25)); draw((-1,0.25)--(-1.25,0.5)); draw((-0.5,0.5)--(-0.25,0.5)); draw((0,0.75)--(0.25,0.5)); draw((-0.75,0)--(-0.5,-0.25)); draw(Circle((1,0),0.25)); draw(((1,0) + 0.25*dir(45))--((1,0) + 0.25*dir(225))); draw(((1,0) + 0.25*dir(135))--((1,0) + 0.25*dir(315))); draw((0.05,-0.9)--(0.05,-1.1)); draw((-0.05,-0.8)--(-0.05,-1.2)); label("$L$", (1.25,0), E); label("$B$", (-0.1,-1.1), SW); label("$S_1$", (-0.5,1.25), NE); label("$S_2$", (-1.25,0.5), SW); label("$S_3$", (-0.5,0.5), SW); label("$S_4$", (0.25,0.5), NE); label("$S_5$", (-0.5,-0.25), SW); [/asy]

Two players play a game as follows: there $n > 1$ rounds and $d \geq 1$ is fixed. In the first round A picks a positive integer $m_1$, then B picks a positive integer $n_1 \not = m_1$. In round $k$ (for $k = 2, \ldots , n$), A picks an integer $m_k$ such that $m_{k-1} < m_k \leq m_{k-1} + d$. Then B picks an integer $n_k$ such that $n_{k-1} < n_k \leq n_{k-1} + d$. A gets $\gcd(m_k,n_{k-1})$ points and B gets $\gcd(m_k,n_k)$ points. After $n$ rounds, A wins if he has at least as many points as B, otherwise he loses. For each $(n, d)$ which player has a winning strategy?

Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.

[u]Bouncy Balls[/u] In the following problems, you will consider the trajectories of balls moving and bouncing off of the boundaries of various containers. The balls are small enough that you can treat them as points. Let us suppose that a ball starts at a point $X$, strikes a boundary (indicated by the line segment $AB$) at $Y$ , and then continues, moving along the ray $Y Z$. Balls always bounce in such a way that $\angle XY A = \angle BY Z$. This is indicated in the above diagram. [img]https://cdn.artofproblemsolving.com/attachments/4/6/42ad28823d839f804d618a1331db43a9ebdca1.png[/img] Balls bounce off of boundaries in the same way light reflects off of mirrors - if the ball hits the boundary at point P, the trajectory after $P$ is the reflection of the trajectory before $P$ through the perpendicular to the boundary at P. A ball inside a rectangular container of width $7$ and height $12$ is launched from the lower-left vertex of the container. It first strikes the right side of the container after traveling a distance of $\sqrt{53}$ (and strikes no other sides between its launch and its impact with the right side). [b]p4.[/b] Find the height at which the ball first contacts the right side. [b]p5.[/b] How many times does the ball bounce before it returns to a vertex? (The final contact with a vertex does not count as a bounce.) Now a ball is launched from a vertex of an equilateral triangle with side length $5$. It strikes the opposite side after traveling a distance of $\sqrt{19}$. [b]p6.[/b] Find the distance from the ball's point of rst contact with a wall to the nearest vertex. [b]p7.[/b] How many times does the ball bounce before it returns to a vertex? (The final contact with a vertex does not count as a bounce.) In this final problem, a ball is again launched from the vertex of an equilateral triangle with side length $5$. [b]p8.[/b] In how many ways can the ball be launched so that it will return again to a vertex for the first time after $2009$ bounces?

William is thinking of an integer between 1 and 50, inclusive. Victor can choose a positive integer $m$ and ask William: "does $m$ divide your number?", to which William must answer truthfully. Victor continues asking these questions until he determines William's number. What is the minimum number of questions that Victor needs to guarantee this?

All sides and diagonals of a $25$-gon are drawn either red or white. Show that at least $500$ triangles, having all three sides are in same color and having all three vertices from the vertices of the $25$-gon, can be found.

Let $n$ be a positive integer. There are $n$ islands with $n-1$ bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let $X$ be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of $X$ over all possible configurations of bridges and island where the fire starts at.

Nair and Yuli play the following game: $1.$ There is a coin to be moved along a horizontal array with $203$ cells. $2.$ At the beginning, the coin is at the first cell, counting from left to right. $3.$ Nair plays first. $4.$ Each of the players, in their turns, can move the coin $1$, $2$, or $3$ cells to the right. $5.$ The winner is the one who reaches the last cell first. What strategy does Nair need to use in order to always win the game?

Find all finite sequences $(x_0, x_1, \ldots,x_n)$ such that for every $j$, $0 \leq j \leq n$, $x_j$ equals the number of times $j$ appears in the sequence.

For a positive integer $n(\geq 2)$, there are $2n$ candies. Alice distributes $2n$ candies into $4n$ boxes $B_1, B_2, \dots, B_{4n}.$ Bob checks the number of candies that Alice puts in each box. After this, Bob chooses exactly $2n$ boxes $B_{k_1}, B_{k_2}, \dots, B_{k_{2n}}$ out of $4n$ boxes that satisfy the following condition, and takes all the candies. (Condition) $k_i - k_{i - 1}$ is either $1$ or $3$ for each $i = 1, 2, \dots, 2n$, and $k_{2n} = 4n$. ($k_0 = 0$) Alice takes all the candies in the $2n$ boxes that Bob did not choose. If Alice and Bob both use their best strategy to take as many candies as possible, how many candies can Alice take?

A student divides all $30$ marbles into $5$ boxes numbered $1, 2, 3, 4, 5$ (after being divided, there may be a box with no marbles). a) How many ways are there to divide marbles into boxes (are two different ways if there is a box with a different number of marbles)? b) After dividing, the student paints those $30$ marbles by a number of colors (each with the same color, one color can be painted for many marbles), so that there are no $2$ marbles in the same box. have the same color and from any $2$ boxes it is impossible to choose $8$ marbles painted in $4$ colors. Prove that for every division, the student must use no less than $10$ colors to paint the marbles. c) Show a division so that with exactly $10$ colors the student can paint the marbles that satisfy the conditions in question b).

Jesse has ten squares, which are labeled $1, 2, \dots, 10$. In how many ways can he color each square either red, green, yellow, or blue such that for all $1 \le i < j \le 10$, if $i$ divides $j$, then the $i$-th and $j$-th squares have different colors?

4. Suppose that four people A, B, C and D decide to play games of tennis doubles. They might first play the team A and B against the team C and D. Next A and C might play B and D. Finally A and D might play B and C. The advantage of this arrangement is that two conditions are satisfied. (a) Each player is on the same team as each other player exactly once. (b) Each player is on the opposing team to each other player exactly twice. Is it possible to arrange a collection of tennis matches satisfying both condition (a) and condition (b) in the following circumstances? (i) There are five players. (ii) There are seven players. (iii) There are nine players.

There is a lamp on each cell of a $2017 \times 2017$ board. Each lamp is either on or off. A lamp is called [i]bad[/i] if it has an even number of neighbours that are on. What is the smallest possible number of bad lamps on such a board? (Two lamps are neighbours if their respective cells share a side.)