Let $k\geq 2$,$n_1,n_2,\cdots ,n_k\in \mathbb{N}_+$,satisfied $n_2|2^{n_1}-1,n_3|2^{n_2}-1,\cdots ,n_k|2^{n_{k-1}}-1,n_1|2^{n_k}-1$. Prove:$n_ 1=n_ 2=\cdots=n_k=1$.

The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$

Let be three positive integers $ a,b,c $ and a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ defined as $$ f(n)=\left\{ \begin{matrix} n-a, & n>c\\ f\left( f(n+b) \right) ,& n\le c \end{matrix} \right. . $$Determine the number of fixed points this function has.

Let be two natural numbers $ m,n, $ and $ m $ pairwise disjoint sets of natural numbers $ A_0,A_1,\ldots ,A_{m-1}, $ each having $ n $ elements, such that no element of $ A_{i\pmod m} $ is divisible by an element of $ A_{i+1\pmod m} , $ for any natural number $ i. $ Determine the number of ordered pairs $$ (a,b)\in\bigcup_{0\le j < m} A_j\times\bigcup_{0\le j < m} A_j $$such that $ a|b $ and such that $ \{ a,b \}\not\in A_k, $ for any $ k\in\{ 0,1,\ldots ,m-1 \} . $ Radu Bumbăcea

Let be a natural number $ n\ge 3. $ Find $$ \inf_{\stackrel{ x_1,x_2,\ldots ,x_n\in\mathbb{R}_{>0}}{1=P\left( x_1,x_2,\ldots ,x_n\right)}}\sum_{i=1}^n\left( \frac{1}{x_i} -x_i \right) , $$where $ P\left( x_1,x_2,\ldots ,x_n\right) :=\sum_{i=1}^n \frac{1}{x_i+n-1} , $ and find in which circumstances this infimum is attained.

Determine the largest natural number $ N $ having the following property: every $ 5\times 5 $ array consisting of pairwise distinct natural numbers from $ 1 $ to $ 25 $ contains a $ 2\times 2 $ subarray of numbers whose sum is, at least, $ N. $ Demetres Christofides and Silouan Brazitikos

Let $AD, BE$, and $CF$ denote the altitudes of triangle $\vartriangle ABC$. Points $E'$ and $F'$ are the reflections of $E$ and $F$ over $AD$, respectively. The lines $BF'$ and $CE'$ intersect at $X$, while the lines $BE'$ and $CF'$ intersect at the point $Y$. Prove that if $H$ is the orthocenter of $\vartriangle ABC$, then the lines $AX, YH$, and $BC$ are concurrent.

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \]Is it possible that both $xy$ and $zt$ are perfect squares?

Prove that there exists an integer $n$, $n\geq 2002$, and $n$ distinct positive integers $a_1,a_2,\ldots,a_n$ such that the number $N= a_1^2a_2^2\cdots a_n^2 - 4(a_1^2+a_2^2+\cdots + a_n^2) $ is a perfect square.

Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i = 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i+1}$ and $ A_iA_{i+2}$ (the addition of indices being mod 3). Let $ B_i, i = 1, 2, 3,$ be the second point of intersection of $ C_{i+1}$ and $ C_{i+2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.

Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

For a natural number $ n, $ a string $ s $ of $ n $ binary digits and a natural number $ k\le n, $ define an $ n,s,k$ -block as a string of $ k $ consecutive elements from $ s. $ We say that two $ n,s,k\text{-blocks} , $ namely, $ a_1a_2\ldots a_k,b_1b_2\ldots b_k, $ are incompatible if there exists an $ i\in\{1,2,\ldots ,k\} $ such that $ a_i\neq b_i. $ Also, for two natural numbers $ r\le n, l, $ we say that $ s $ is $ r,l $ -typed if there are, at most, $ l $ pairwise incompatible $ n,s,r\text{-blocks} . $ Let be a $ 3,7\text{-typed} $ string $ t $ consisting of $ 10000 $ binary digits. Determine the maximum number $ M $ that satisfies the condition that $ t $ is $ 10,M\text{-typed} . $ Cătălin Gherghe

Let $ I,O $ denote the incenter, respectively, the circumcenter of a triangle $ ABC. $ The $ A\text{-excircle} $ touches the lines $ AB,AC,BC $ at $ K,L, $ respectively, $ M. $ The midpoint of $ KL $ lies on the circumcircle of $ ABC. $ Show that the points $ I,M,O $ are collinear. Павел Кожевников

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

Alice and Bob play the following game. To start, Alice arranges the numbers $1,2,\ldots,n$ in some order in a row and then Bob chooses one of the numbers and places a pebble on it. A player's turn consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the number $k$ at most $k$ times. The two players alternate taking turns beginning with Alice. The first player who cannot make a move loses. For each positive integer $n$, determine who has a winning strategy.

Determine the largest value the expression $$ \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j} $$may achieve, as $ x_1,x_2,x_3,x_4 $ run through the non-negative real numbers, and add up to $ 1. $ Find also the specific values of this numbers that make the above sum achieve the asked maximum.

Let $ABC$ be an acute triangle with $AB<BC$. Let $I$ be the incenter of $ABC$, and let $\omega$ be the circumcircle of $ABC$. The incircle of $ABC$ is tangent to the side $BC$ at $K$. The line $AK$ meets $\omega$ again at $T$. Let $M$ be the midpoint of the side $BC$, and let $N$ be the midpoint of the arc $BAC$ of $\omega$. The segment $NT$ intersects the circumcircle of $BIC$ at $P$. Prove that $PM\parallel AK$.

Given an integer $n\geq 2,$ colour red exactly $n$ cells of an infinite sheet of grid paper. A rectangular grid array is called special if it contains at least two red opposite corner cells; single red cells and 1-row or 1-column grid arrays whose end-cells are both red are special. Given a configuration of exactly $n$ red cells, let $N$ be the largest number of red cells a special rectangular grid array may contain. Determine the least value $N$ may take over all possible configurations of exactly $n$ red cells