Point $C_{1}$ of hypothenuse $AC$ of a right-angled triangle $ABC$ is such that $BC = CC_{1}$. Point $C_{2}$ on cathetus $AB$ is such that $AC_{2} = AC_{1}$; point $A_{2}$ is defined similarly. Find angle $AMC$, where $M$ is the midpoint of $A_{2}C_{2}$.

Integers $1 \le n \le 200$ are written on a blackboard just one by one. We surrounded just $100$ integers with circle. We call a square of the sum of surrounded integers minus the sum of not surrounded integers $score$ of this situation. Calculate the average score in all ways.

Find all pairs of positive integers $(a,b)$ such that $\sqrt{a-1}+\sqrt{b-1}=\sqrt{ab-1}.$

A triangle is given, all the angles of which are smaller than $\phi$, where $\phi <2\pi / 3$. Prove that in space there is a point from which all sides of the triangle are visible at an angle $\phi$.

In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's [i]score[/i] is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n\times n$ grid with probability $k$; he notices that the expected value of the score of the resulting grid is equal to $k$, too! Given that $k > 0.9999$, find the minimum possible value of $n$. [i]Proposed by Andrew Wu[/i]

Let $u_{jk}$ be a real number for each $j=1,2,3$ and each $k=1,2$ and let $N$ be an integer such that \[\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N\] Let $M$ and $l$ be positive integers such that $l^2 <(M+1)^3$. Prove that there exist integers $\xi_1,\xi_2,\xi_3$ not all zero, such that \[\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}\]

Consider the segment $[0; 1]$. At each step we may split one of the available segments into two new segments and write the product of lengths of these two new segments onto a blackboard. Prove that the sum of the numbers on the blackboard never will exceed $1/2$. [i]Mikhail Lukin[/i]

Kristoff is planning to transport a number of indivisible ice blocks with positive integer weights from the north mountain to Arendelle. He knows that when he reaches Arendelle, Princess Anna and Queen Elsa will name an ordered pair $(p,q)$ of nonnegative integers satisfying $p + q \le 2016$. Kristoff must then give Princess Anna \emph{exactly} $p$ kilograms of ice. Afterward, he must give Queen Elsa $\emph{exactly}$ $q$ kilograms of ice. What is the minimum number of blocks of ice Kristoff must carry to guarantee that he can always meet Anna and Elsa's demands, regardless of which $p$ and $q$ are chosen?

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

Find all functions $u : R \to R$ for which there is a strictly increasing function $f : R \to R$ such that $f(x+y) = f(x)u(y)+ f(y)$ for all $x,y \in R$.

Let $A$ be the set of positive integers of the form $a^2 +2b^2$, where $a$ and $b$ are integers and $b \neq 0$. Show that if $p$ is a prime number and $p^2 \in A$, then $p \in A$.

Compute the smallest positive integer such that, no matter how you rearrange its digits (in base ten), the resulting number is a multiple of $63.$

There are There are $64$ towns in a country, and some pairs of towns are connected by roads but we do not know these pairs. We may choose any pair of towns and find out whether they are connected by a road. Our aim is to determine whether it is possible to travel between any two towns using roads. Prove that there is no algorithm which would enable us to do this in less than $2016$ questions. but we do not know these pairs. We may choose any pair of towns and find out whether they are connected by a road. Our aim is to determine whether it is possible to travel between any two towns using roads. Prove that there is no algorithm which would enable us to do this in less than $2016$ questions.

Let $z = \cos \frac{2\pi}{2011} + i\sin \frac{2\pi}{2011}$, and let \[ P(x) = x^{2008} + 3x^{2007} + 6x^{2006} + \cdots + \frac{2008 \cdot 2009}{2} x + \frac{2009 \cdot 2010}{2} \] for all complex numbers $x$. Evaluate $P(z)P(z^2)P(z^3) \cdots P(z^{2010})$.

Let $m, k$, and $c$ be positive integers with $k > c$, and let $\lambda$ be a positive, non-integer real root of the equation $\lambda^{m+1} - k \lambda^m - c = 0$. Let $f : Z^+ \to Z$ be defined by $f(n) = \lfloor \lambda n \rfloor$ for all $n \in Z^+$. Show that $f^{m+1}(n) \equiv cn - 1$ (mod $k$) for all $n \in Z^+$. (Here, $Z^+$ denotes the set of positive integers, $ \lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $f^{m+1}(n) = f(f(... f(n)...))$ where $f$ appears $m + 1$ times.)

A lattice point is defined as a point on the plane with integer coordinates. Show that for all positive integers $n$, there is a circle on the plane with exactly n lattice points in its interior (not including its boundary).

Let $a,b,c,d \in [0,1] .$ Prove that$$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+d}+\frac{1}{1+d+a}\leq \frac{4}{1+2\sqrt[4]{abcd}}$$

Let $ABC$ be a triangle with incenter $I$. Let $U$, $V$ and $W$ be the intersections of the angle bisectors of angles $A$, $B$, and $C$ with the incircle, so that $V$ lies between $B$ and $I$, and similarly with $U$ and $W$. Let $X$, $Y$, and $Z$ be the points of tangency of the incircle of triangle $ABC$ with $BC$, $AC$, and $AB$, respectively. Let triangle $UVW$ be the [i]David Yang triangle[/i] of $ABC$ and let $XYZ$ be the [i]Scott Wu triangle[/i] of $ABC$. Prove that the David Yang and Scott Wu triangles of a triangle are congruent if and only if $ABC$ is equilateral. [i]Proposed by Owen Goff[/i]

Let $a_1, . . . , a_{2020}$ be a sequence of real numbers such that $a_1 = 2^{-2019}$, and $a^2_{n-1}a_n = a_n-a_{n-1}$. Prove that $a_{2020} <\frac{1}{2^{2019} -1}$

In a classroom there are $m$ students. During the month of July each of them visited the library at least once but none of them visited the library twice in the same day. It turned out that during the month of July each student visited the library a different number of times, furthermore for any two students $A$ and $B$ there was a day in which $A$ visited the library and $B$ did not and there was also a day when $B$ visited the library and $A$ did not do so. Determine the largest possible value of $m$.

Calculate $\sin^3 a + \cos^3 a$ if you know that $\sin a+ \cos a = m$.

Prove that for all positive integers $n,m$ and all real numbers $x, y > 0$ the following inequality holds: \[(n - 1)(m- 1)(x^{n+m} + y^{n+m}) + (n + m - 1)(x^ny^m + x^my^n)\\ \\ \ge nm(x^{n+m-1}y + xy^{n+m-1}).\]

Find the number of ways to put a number in every unit square of a $3 \times 3$ square such that any number is divisible by the number directly to the top and the number directly to the left of it, and the top-left number is $1$ and the bottom right number is $2013$.

An octomino is made by joining $8$ congruent squares edge to edge. Three examples are shown below. How many octominoes have at least $2$ lines of symmetry? [asy] size(8cm,0); filldraw((0,1)--(0,2)--(1,2)--(1,1)--cycle,grey); filldraw((0,2)--(0,3)--(1,3)--(1,2)--cycle,grey); filldraw((0,3)--(0,4)--(1,4)--(1,3)--cycle,grey); filldraw((1,0)--(1,1)--(2,1)--(2,0)--cycle,grey); filldraw((1,1)--(1,2)--(2,2)--(2,1)--cycle,grey); filldraw((1,2)--(1,3)--(2,3)--(2,2)--cycle,grey); filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle,grey); filldraw((2,2)--(2,3)--(3,3)--(3,2)--cycle,grey); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle,grey); filldraw((4,1)--(4,2)--(5,2)--(5,1)--cycle,grey); filldraw((4,2)--(4,3)--(5,3)--(5,2)--cycle,grey); filldraw((4,3)--(4,4)--(5,4)--(5,3)--cycle,grey); filldraw((5,1)--(5,2)--(6,2)--(6,1)--cycle,grey); filldraw((5,3)--(5,4)--(6,4)--(6,3)--cycle,grey); filldraw((6,2)--(6,3)--(7,3)--(7,2)--cycle,grey); filldraw((6,3)--(6,4)--(7,4)--(7,3)--cycle,grey); filldraw((8,3)--(8,4)--(9,4)--(9,3)--cycle,grey); filldraw((9,3)--(9,4)--(10,4)--(10,3)--cycle,grey); filldraw((10,3)--(10,4)--(11,4)--(11,3)--cycle,grey); filldraw((11,2)--(11,3)--(12,3)--(12,2)--cycle,grey); filldraw((11,3)--(11,4)--(12,4)--(12,3)--cycle,grey); filldraw((12,2)--(12,3)--(13,3)--(13,2)--cycle,grey); filldraw((12,3)--(12,4)--(13,4)--(13,3)--cycle,grey); filldraw((13,3)--(13,4)--(14,4)--(14,3)--cycle,grey); [/asy]

Let $p = (a_{1}, a_{2}, \ldots , a_{12})$ be a permutation of $1, 2, \ldots, 12$. We will denote \[S_{p} = |a_{1}-a_{2}|+|a_{2}-a_{3}|+\ldots+|a_{11}-a_{12}|\]We'll call $p$ $\textit{optimistic}$ if $a_{i} > \min(a_{i-1}, a_{i+1})$ $\forall i = 2, \ldots, 11$. $a)$ What is the maximum possible value of $S_{p}$. How many permutations $p$ achieve this maximum?$\newline$ $b)$ What is the number of $\textit{optimistic}$ permtations $p$? $c)$ What is the maximum possible value of $S_{p}$ for an $\textit{optimistic}$ $p$? How many $\textit{optimistic}$ permutations $p$ achieve this maximum?