Let $X$ and $Y$ be two sets of points in the plane and $M$ be a set of segments connecting points from $X$ and $Y$ . Let $k$ be a natural number. Prove that the segments from $M$ can be painted using $k$ colors in such a way that for any point $x \in X \cup Y$ and two colors $\alpha$ and $\beta$ $(\alpha \neq \beta)$, the difference between the number of $\alpha$-colored segments and the number of $\beta$-colored segments originating in $X$ is less than or equal to $1$.

Given regular hexagon $ABCDEF$, compute the probability that a randomly chosen point inside the hexagon is inside triangle $PQR$, where $P$ is the midpoint of $AB$, $Q$ is the midpoint of $CD$, and $R$ is the midpoint of $EF$.

The diagram shows some squares whose sides intersect other squares at the midpoints of their sides. The shaded region has total area $7$. Find the area of the largest square. [img]https://cdn.artofproblemsolving.com/attachments/3/a/c3317eefe9b0193ca15f36599be3f6c22bb099.png[/img]

Let $q$ be a positive rational number. Two ants are initially at the same point $X$ in the plane. In the $n$-th minute $(n = 1,2,...)$ each of them chooses whether to walk due north, east, south or west and then walks the distance of $q^n$ metres. After a whole number of minutes, they are at the same point in the plane (not necessarily $X$), but have not taken exactly the same route within that time. Determine all possible values of $q$. Proposed by Jeremy King, UK

Find all integers \( n > 1 \) such that all prime divisors of \( n^6 - 1 \) divide \( (n^2 - 1)(n^3 - 1) \).

A bored student walks down a hallway where there is a row of closed lockers, numbered from $1$ to $1024$. Opens cabinet No. $1$, then skips one cabinet and opens the next, and so on successively. When he reaches the end of the row, he turns around and starts again: he opens the first cabinet it finds closed, he skips the next closed cabinet and so on until the start from the hallway. goes from beginning to end, from end to beginning of the corridor until all the cabinets are left open. What is the number of the last cabinet he opened?

Let $x,y,z$ be pairwise coprime positive integers and $p\ge5$ and $q$ be prime numbers which satisfy the following conditions: (i) $6p$ does not divide $q-1$; (ii) $q$ divides $x^2+xy+y^2$; (iii) $q$ does not divide $x+y-z$. Prove that $x^p+y^p\ne z^p$.

Let $M_1, M_2, . . ., M_{11}$ be $5-$element sets such that $M_i \cap M_j \neq {\O}$ for all $i, j \in \{1, . . ., 11\}$. Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection.

Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.

The Den has two deals on chicken wings. The first deal is $4$ chicken wings for $3$ dollars, and the second deal is $11$ chicken wings for $ 8$ dollars. If Jeremy has $18$ dollars, what is the largest number of chicken wings he can buy?

Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. Show that $$\frac{1 - a^2}{a + bc} + \frac{1 - b^2}{b + ca} + \frac{1 - c^2}{c + ab} \ge 6$$

Define $x\star y$ to be $xy\cdot \min(x,y)$ and $x\diamond y$ to be $xy\cdot \max(x,y)$. Suppose $ab=4$. Find the value of \[ (a\star b)\cdot (a\diamond b). \]

From a set containing $6$ positive and consecutive integers they are extracted, randomly and with replacement, three numbers $a, b, c$. Determine the probability that even $a^b + c$ generates as a result .

Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.

Given $x_0\in\mathbb R$, $f,g:\mathbb R\to\mathbb R$, we define the $\emph{non-redundant binary tree}$ $T(x_0,f,g)$ in the following way: [list=1] [*]The tree $T$ initially consists of just $x_0$ at height $0$. [*]Let $v_0,\dots,v_k$ be the vertices at height $h$. Then the vertices of height $h+1$ are added to $T$ by: for $i=0,1,\dots,k$, $f(v_i)$ is added as a child of $v_i$ if $f(v_i)\not\in T$, and $g(v_i)$ is added as a child of $v_i$ if $g(v_i)\not\in T$. [/list] For example, if $f(x)=x+1$ and $g(x)=x-1$, then the first three layers of $T(0,f,g)$ look like: [asy] size(100); draw((-0.1,-0.2)--(-0.4,-0.8),EndArrow(size=3)); draw((0.1,-0.2)--(0.4,-0.8),EndArrow(size=3)); draw((-0.6,-1.2)--(-0.9,-1.8),EndArrow(size=3)); draw((0.6,-1.2)--(0.9,-1.8),EndArrow(size=3)); label("$0$",(0,0)); label("$1$",(-.5,-1)); label("$-1$",(.5,-1)); label("$2$",(-1,-2)); label("$-2$",(1,-2));[/asy] If $f(x)=1024x-2047\lfloor x/2\rfloor$ and $g(x)=2x-3\lfloor x/2\rfloor+2\lfloor x/4\rfloor$, then how many vertices are in $T(2016,f,g)$?

Define the sequence $\{a_n\}_{n\geq 1}$ as $a_n=n-1$, $n\leq 2$ and $a_n=$ remainder left by $a_{n-1}+a_{n-2}$ when divided by $3$ $\forall n\geq 2$. Then $\sum_{i=2018}^{2025}a_i=$? (A)$6$ (B)$7$ (C)$8$ (D)$9$

Let $n$ be a positive integer such that $n(n+3)$ is a perfect square of an integer, prove that $n$ is not a multiple of $3$.

The vertices of a regular $2n$-gon ($n \ge 3$) are labelled with the numbers $1,2,...,2n$ so that the sum of the numbers at any two adjacent vertices equals the sum of the numbers at the vertices diametrically opposite to them. Show that this is only possible if $n$ is odd.

Which statement is correct? $ \textbf{(A)}\ \text{If } x<0, \text{ then } x^2 > x. \qquad \textbf{(B)}\ \text{If } x^2 > 0, \text{ then } x > 0. \qquad$ $ \textbf{(C)}\ \text{If } x^2 > x, \text{ then } x>0. \qquad \textbf{(D)}\ \text{If } x^2 > x, \text{ then } x<0. \qquad$ $ \textbf{(E)}\ \text{If } x<1, \text{ then } x^2<x.$

For real variables $0 \leq x, y, z, w \leq 1$, find the maximum value of $$x(1-y)+2y(1-z)+3z(1-w)+4w(1-x)$$

Let $ \mathbb{Z}$ denote the set of all integers. Prove that for any integers $ A$ and $ B,$ one can find an integer $ C$ for which $ M_1 \equal{} \{x^2 \plus{} Ax \plus{} B : x \in \mathbb{Z}\}$ and $ M_2 \equal{} {2x^2 \plus{} 2x \plus{} C : x \in \mathbb{Z}}$ do not intersect.

Prove that in any parallelepiped the sum of the lengths of the edges is less than or equal to twice the sum of the lengths of the four diagonals.

How many of the twelve pentominoes pictured below have at least one line of symmetry? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$ [asy]unitsize(5mm); defaultpen(linewidth(1pt)); draw(shift(2,0)*unitsquare); draw(shift(2,1)*unitsquare); draw(shift(2,2)*unitsquare); draw(shift(1,2)*unitsquare); draw(shift(0,2)*unitsquare); draw(shift(2,4)*unitsquare); draw(shift(2,5)*unitsquare); draw(shift(2,6)*unitsquare); draw(shift(1,5)*unitsquare); draw(shift(0,5)*unitsquare); draw(shift(4,8)*unitsquare); draw(shift(3,8)*unitsquare); draw(shift(2,8)*unitsquare); draw(shift(1,8)*unitsquare); draw(shift(0,8)*unitsquare); draw(shift(6,8)*unitsquare); draw(shift(7,8)*unitsquare); draw(shift(8,8)*unitsquare); draw(shift(9,8)*unitsquare); draw(shift(9,9)*unitsquare); draw(shift(6,5)*unitsquare); draw(shift(7,5)*unitsquare); draw(shift(8,5)*unitsquare); draw(shift(7,6)*unitsquare); draw(shift(7,4)*unitsquare); draw(shift(6,1)*unitsquare); draw(shift(7,1)*unitsquare); draw(shift(8,1)*unitsquare); draw(shift(6,0)*unitsquare); draw(shift(7,2)*unitsquare); draw(shift(11,8)*unitsquare); draw(shift(12,8)*unitsquare); draw(shift(13,8)*unitsquare); draw(shift(14,8)*unitsquare); draw(shift(13,9)*unitsquare); draw(shift(11,5)*unitsquare); draw(shift(12,5)*unitsquare); draw(shift(13,5)*unitsquare); draw(shift(11,6)*unitsquare); draw(shift(13,4)*unitsquare); draw(shift(11,1)*unitsquare); draw(shift(12,1)*unitsquare); draw(shift(13,1)*unitsquare); draw(shift(13,2)*unitsquare); draw(shift(14,2)*unitsquare); draw(shift(16,8)*unitsquare); draw(shift(17,8)*unitsquare); draw(shift(18,8)*unitsquare); draw(shift(17,9)*unitsquare); draw(shift(18,9)*unitsquare); draw(shift(16,5)*unitsquare); draw(shift(17,6)*unitsquare); draw(shift(18,5)*unitsquare); draw(shift(16,6)*unitsquare); draw(shift(18,6)*unitsquare); draw(shift(16,0)*unitsquare); draw(shift(17,0)*unitsquare); draw(shift(17,1)*unitsquare); draw(shift(18,1)*unitsquare); draw(shift(18,2)*unitsquare);[/asy]

In a school there are $40$ different clubs, each of them contains exactly $30$ children. For every $i$ from $1$ to $30$ define $n_i$ as a number of children who attend exactly $i$ clubs. Prove that it is possible to organize $40$ new clubs with $30$ children in each of them such, that the analogical numbers $n_1, n_2,..., n_{30}$ will be the same for them.

Let $x, y$ be two non-negative integers. Prove that $47$ divides $3^x - 2^y$ if and only if $23$ divides $4x + y$.