Let $O$ be the circumcenter of acutangle triangle $ABC$ and let $A_1$ be some point in the smallest arc $BC$ of the circumcircle of $ABC$. Let $A_2$ and $A_3$ points on sides $AB$ and $AC$, respectively, such that $\angle BA_1A_2 = \angle OAC$ and $\angle CA_1A_3 = \angle OAB$. Prove that the line $A_2A_3$ passes through the orthocenter of $ABC$.

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ a function having the property that $$ \left| f(x+y)+\sin x+\sin y \right|\le 2, $$ for all real numbers $ x,y. $ [b]a)[/b] Prove that $ \left| f(x) \right|\le 1+\cos x, $ for all real numbers $ x. $ [b]b)[/b] Give an example of what $ f $ may be, if the interval $ \left( -\pi ,\pi \right) $ is included in its [url=https://en.wikipedia.org/wiki/Support_(mathematics)]support.[/url]

Points $M$ and $N$ are the midpoints of the sides $BC$ and $AD$, respectively, of a convex quadrilateral $ABCD$. Is it possible that $$ AB+CD>\max(AM+DM,BN+CN)? $$ [i](Folklore)[/i]

Find, with proof, all ordered pairs of positive integers $(a, b)$ with the following property: there exist positive integers $r$, $s$, and $t$ such that for all $n$ for which both sides are defined, [center]${{n\choose{a}}\choose{b}}=r {{n+s}\choose{t}}$ .[/center]

Determine all functions $f : R \to R$ such that for all $x, y\in R$ holds $$f (f(x) + 2f(y)) = f(2x) + 8y + 6.$$

In a triangle $ABC, X$ and $Y$ are points on the segments $AB$ and $AC$, respectively, such that $AX : XB = 1 : 2$ and $AY :YC = 2:1$. If the area of triangle $AXY$ is $10$, then what is the area of triangle $ABC$?

Each of $N$ people chooses a random integer number between $1$ and $19$ (including $1$ and $19$, and not necessarily with the same distribution). The random numbers chosen by the people are independent from each other, and it is true that each person chooses each of the $19$ numbers with probability at most $99\%$. They add up the $N$ chosen numbers, and take the remainder of the sum divided by $19$. Prove that the distribution of the result tends to the uniform distribution exponentially, i.e. there exists a number $0<c<1$ such that the mod $19$ remainder of the sum of the $N$ chosen numbers equals each of the mod $19$ remainders with probability between $\frac{1}{19}-c^{N}$ and $\frac{1}{19}+c^{N}$.

Find the greatest common divisor of the numbers \[ 2^{561}\minus{}2, 3^{561}\minus{}3, \ldots, 561^{561}\minus{}561.\]

Let $n\in\mathbb{N}^*$. Solve the equation $\sum_{k=0}^n C_n^k\cos2kx=\cos nx$ in $\mathbb{R}$.

Let $F(z)=\frac{z+i}{z-i}$ for all complex numbers $z\not= i,$ and let $z_n=F(z_{n-1})$ for all positive integers $n.$ Given that $z_0=\frac 1{137}+i$ and $z_{2002}=a+bi,$ where $a$ and $b$ are real numbers, find $a+b.$

A unit square $ABCD$ is balanced on a flat table with only its vertex $A$ touching the table, such that $AC$ is perpendicular to the table. The square loses balance and falls to one side. At the end of the fall, $A$ is in the same place as before, and $B$ is also touching the table. Compute the area swept by the square during its fall.

What is the greatest positive integer $x$ for which $2^{2^x+1}+2$ is divisible by $17$?

Prove that $\sin x + \frac12 \sin 2x + \frac13 \sin 3x > 0$ for $0 < x < 180^o$.

Zeros and ones are placed in each square of a rectangular board. Such a board is said to be [i]varied[/i] if each row contains at least one $0$ and at least two $1$s. Given n$\geq 3,$ find all integers $k>1$ with the following property: The columns of each varied board of $k$ rows and n columns can be permuted so that in each row of the new board the $1$s do not form a block (that is, there are at least two $1$s that are separated by one or more $0$s).

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?

The sidelengths of a polygon with $1994$ sides are $a_{i}=\sqrt{i^2 +4}$ $ \; (i=1,2,\cdots,1994)$. Prove that its vertices are not all on lattice points.

In a triangle $ABC$, let $P$ be a point on the bisector of $\angle BAC$ and let $A',B'$ and $C'$ be points on lines $BC,CA$ and $AB$ respectively such that $PA'$ is perpendicular to $BC,PB'\perp AC$, and $PC'\perp AB$. Prove that $PA'$ and $B'C'$ intersect on the median $AM$, where $M$ is the midpoint of $BC$.

Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]

In triangle $ABC$ the median $BM$ is equal to half of the side $BC$. Show that $\angle ABM = \angle BCA + \angle BAC$. [i](Proposed by Anton Trygub)[/i]

Let $ABC$ be an acute triangle and let $MNPQ$ be a square inscribed in the triangle such that $M ,N \in BC$, $P \in AC$, $Q \in AB$. Prove that $area \, [MNPQ] \le \frac12 area\, [ABC]$.

We have tiles (which are build from squares of side length 1) of following shapes: [asy] unitsize(0.5 cm); draw((1,0)--(2,0)); draw((1,1)--(2,1)); draw((1,0)--(1,1)); draw((2,0)--(2,1)); draw((0,1)--(1,1)); draw((0,2)--(1,2)); draw((0,1)--(0,2)); draw((1,1)--(1,2)); draw((0, 0)--(1, 0)); draw((0, 0)--(0, 1)); draw((5,0)--(6,0)); draw((5,1)--(6,1)); draw((5,0)--(5,1)); draw((6,0)--(6,1)); draw((4,1)--(5,1)); draw((5,2)--(6,2)); draw((5,1)--(5,2)); draw((6,1)--(6,2)); draw((4, 0)--(5, 0)); draw((4, 0)--(4, 1)); draw((6,2)--(7,2)); draw((7,1)--(7,2)); draw((6,1)--(7,1)); draw((11,0)--(12,0)); draw((11,1)--(12,1)); draw((11,0)--(11,1)); draw((12,0)--(12,1)); draw((10,1)--(11,1)); draw((10,2)--(11,2)); draw((10,1)--(10,2)); draw((11,1)--(11,2)); draw((10, 0)--(11, 0)); draw((10, 0)--(10, 1)); draw((9, 2)--(9, 1)); draw((9,1)--(10, 1)); draw((9,2)--(10,2)); [/asy] For each odd integer $n \ge 7$, determine minimal number of these tiles needed to arrange square with side of length $n$. (Attention: Tiles can be rotated, but they can't overlap.)

Consider three circles $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$, with centres $O_1$, $O_2$ and $O_3$, respectively, such that each pair of circles is externally tangent. Suppose we have another circle $\Gamma$ with centre $O$ on the line segment $O_1O_3$ such that $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ are each internally tangent to $\Gamma$. Show that $\angle O_1O_2O_3$ measures less than $90^\circ$.

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m- n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$. [i]Proposed by Mahyar Sefidgaran, Iran[/i]

In how many ways can we construct a square with dimensions $4\times 4$ using $4$ white, $4$ green , $4$ red and 4 $blue$ squares of dimensions $1\times 1$, such that in every horizontal and in every certical line, squares have different colours .

In a rectangular array of nonnegative reals with $m$ rows and $n$ columns, each row and each column contains at least one positive element. Moreover, if a row and a column intersect in a positive element, then the sums of their elements are the same. Prove that $m=n$.