We attach to the vertices of a regular hexagon the numbers $1$, $0$, $0$, $0$, $0$, $0$. Now, we are allowed to transform the numbers by the following rules: (a) We can add an arbitrary integer to the numbers at two opposite vertices. (b) We can add an arbitrary integer to the numbers at three vertices forming an equilateral triangle. (c) We can subtract an integer $t$ from one of the six numbers and simultaneously add $t$ to the two neighbouring numbers. Can we, just by acting several times according to these rules, get a cyclic permutation of the initial numbers? (I. e., we started with $1$, $0$, $0$, $0$, $0$, $0$; can we now get $0$, $1$, $0$, $0$, $0$, $0$, or $0$, $0$, $1$, $0$, $0$, $0$, or $0$, $0$, $0$, $1$, $0$, $0$, or $0$, $0$, $0$, $0$, $1$, $0$, or $0$, $0$, $0$, $0$, $0$, $1$ ?)

Haydn picks two different integers between $1$ and $100$, inclusive, uniformly at random. The probability that their product is divisible by $4$ can be expressed in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Suppose that $ABCD$ is a rectangle with sides of length $12$ and $18$. Let $S$ be the region of points contained in $ABCD$ which are closer to the center of the rectangle than to any of its vertices. Find the area of $S$.

Of six cities $S_1,S_2,...,S_6$ and two airlines $A$ and $B$ it is given that for every pair $(S_i,S_j)$ (where $i \ne j$) exactly one of the airlines has a connection from $S_i$ to $S_j$ and maintains back. (a) Prove that the air net of one of the companies contains a triangle. (b) Prove that in the two air nets there are even two triangles. [hide=]original wording]Van zes steden $S_1,S_2,...,S_6$ en twee luchtvaartmaatschappijen $A$ en $B$ is gegeven, dat voor ieder paar $(S_i,S_j)$ (waar $i \ne j$) precies één van de maatschappijen een verbinding van $S_i$ naar $S_j$ en terug onderhoudt, (a) Bewijs, dat het luchtnet van één van de maaschappijen een driehoek bevat; (b) Bewijs, dat er in de twee luchtnetten zelfs twee driehoeken zijn.[/hide]

Consider all points with integer coordinates in the carthesian plane. If one draws a circle with M(0,0) and a well-chose radius r, the circles goes through some of those points. (like circle with $r=2\sqrt2$ goes through 4 points) Prove that $\forall n\in \mathbb{N}, \exists r$ so that the circle with midpoint 0,0 and radius $r$ goes through at least $n$ points.

For $H\subset \mathbb Z$ and $n\in\mathbb Z$ let $h_n$ denote the number of finite subsets of $H$ in which the sum of the elements is $n$. Determine whether there exists $H\subset \mathbb Z$ for which $0\notin H$ and $h_n$ is a finite even number for every $n\in\mathbb{Z}$. (The sum of the elements of the empty set is $0$.) [i]Proposed by Csongor Beke, Cambridge[/i]

Find all functions $f: \mathbb{N} \rightarrow [0, +\infty)$ such that $f(1000)=10$ and $f(n+1)= \sum_{k=1}^n \frac{1}{f^2(k) + f(k)f(k+1) + f^2(k+1)}$ for all $n \in \mathbb{N}$. (Here, $f^2(i)$ means $(f(i))^2$.)

In a 2025 by 2025 grid, every cell initially contains a `1'. Every minute, we simultaneously replace the number in each cell with the sum of numbers in the cells that share an edge with it. (For example, after the first minute, the number 2 is written in each of the four corner cells.) After 2025 minutes, we colour the board in checkerboard fashion, such that the top left corner is black. Find the difference between the sum of numbers in black cells and the sum of numbers in white cells. [i]Proposed by chorn[/i]

Let $I$, $O$, $H$, and $\Omega$ be the incenter, circumcenter, orthocenter, and the circumcircle of the triangle $ABC$, respectively. Assume that line $AI$ intersects with $\Omega$ again at point $M\neq A$, line $IH$ and $BC$ meets at point $D$, and line $MD$ intersects with $\Omega$ again at point $E\neq M$. Prove that line $OI$ is tangent to the circumcircle of triangle $IHE$. [i]Proposed by Li4 and Leo Chang.[/i]

Let $A,B\in \mathcal{M}_n(\mathbb{R})$ such that $B^2=I_n$ and $A^2=AB+I_n$. Prove that: \[\det A\le \left(\frac{1+\sqrt{5}}{2}\right)^n\]

Let $K$ be the point of intersection of $AB$ and the line touching the circumcircle of $\triangle ABC$ at $C$ where $m(\widehat {A}) > m(\widehat {B})$. Let $L$ be a point on $[BC]$ such that $m(\widehat{ALB})=m(\widehat{CAK})$, $5|LC|=4|BL|$, and $|KC|=12$. What is $|AK|$? $ \textbf{(A)}\ 4\sqrt 2 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ \text{None of the above} $

Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have \[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \] Find the number of [i]good[/i] functions.

Let $ABC$ be a triangle such that $AB <AC$, $\omega$ its inscribed circle and $\Gamma$ its circumscribed circle. Let also $\omega_b$ be the excircle relative to vertex $B$, then $B'$ is the point of tangency between $\omega_b$ and $(AC)$. Similarly, let the circle $\omega_c$ be the excircle exinscribed relative to vertex $C$, then $C'$ is the point of tangency between $\omega_c$ and $(AB)$. Finally, let $I$ be the center of $\omega$ and $X$ the point of $\Gamma$ such that $\angle XAI$ is a right angle. Prove that the triangles $XBC'$ and $XCB'$ are congruent.

A cyclic quadrilateral is divided into four quadrilaterals by two lines passing through its inner point. Three of these quadrilaterals are cyclic with equal circumradii. Prove that the fourth part also is cyclic quadrilateral and its circumradius is the same. (A.Blinkov)

Three numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3$, with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Let $ABC$ be a triangle with $AB = BC$. Prove that $\triangle ABC$ is an obtuse triangle if and only if the equation $$Ax^2 + Bx + C = 0$$ has two distinct real roots, where $A$, $B$, $C$, are the angles in radians.

Triangle $ABC$ has circumcircle $\omega$. The angle bisectors of $\angle A$ and $\angle B$ intersect $\omega$ at points $D$ and $E$ respectively. $DE$ intersects $BC$ and $AC$ at $X$ and $Y$ respectively. Given $DX = 7,$ $XY = 8$ and $YE = 9,$ the area of $\triangle ABC$ can be written as $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are positive integers, $\gcd(a,c) = 1,$ and $b$ is square free. Find $a+b+c.$

Let be $a,b,c$ complex numbers such that $|a|=|b|=|c|=r$ then show that $\left | \frac{ab+bc+ca}{a+b+c}\right|=r$

$x^3+(\log_2{5}+\log_3{2}+\log_5{3})x=(\log_2{3}+\log_3{5}+\log_5{2})x^2+1$

For each integer $n > 3$ find all quadruples $(n_1,n_2,n_3,n_4)$ of positive integers with $n_1 +n_2 +n_3 +n_4 = n$ which maximize the expression $$\frac{n!}{n_1!n_2!n_3!n_4!}2^{ {n_1 \choose 2}+{n_2 \choose 2}+{n_3 \choose 2}+{n_4 \choose 2}+n_1n_2+n_2n_3+n_3n_4}$$

A circle with center $O$ is inscribed in an angle. Let $A$ be the reflection of $O$ across one side of the angle. Tangents to the circle from $A$ intersect the other side of the angle at points $B$ and $C$. Prove that the circumcenter of triangle $ABC$ lies on the bisector of the original angle. (I.Sharygin)

Each interior point of an equilateral triangle with sides equal to $1$ lies in one of six circles of the same radius $ r $. Prove that $ r \geq \frac {{\sqrt 3}} {{10}} $.

A beetle crawls along the edges of an $n$-lateral pyramid, starting and ending at the midpoint $A$ of a base edge and passing through each point at most once. How many ways are there for the beetle to do this (two ways are said to be equal if they go through the same vertices)? Show that the sum of the numbers of passed vertices (over all these ways) equals $1^2 +2^2 +\ldots +n^2. $

Find all natural numbers that can be written in the form $\frac{4ab}{ab^2+1}$ for some natural $a,b$. (nooonuii)

Let $k$ be a positive integer and $M_k$ the set of all the integers that are between $2 \cdot k^2 + k$ and $2 \cdot k^2 + 3 \cdot k,$ both included. Is it possible to partition $M_k$ into 2 subsets $A$ and $B$ such that \[ \sum_{x \in A} x^2 = \sum_{x \in B} x^2. \]