In a triangle $ABC$, $\angle ABC= 60^o$, point $I$ is the incenter. Let the points $P$ and $T$ on the sides $AB$ and $BC$ respectively such that $PI \parallel BC$ and $TI \parallel AB$ , and points $P_1$ and $T_1$ on the sides $AB$ and $BC$ respectively such that $AP_1 = BP$ and $CT_1 = BT$. Prove that point $I$ lies on segment $P_1T_1$. (Anton Trygub)

There is a regular hexagon that is cut direct to $6n^2$ equilateral triangles (Fig.). There are arranged $2n$ rooks, neither of which beats each other (the rooks hit in directions parallel to sides of the hexagon). Prove that if we consider chess coloring all $6n^2$ equilateral triangles, then the number of rooks that stand on black triangles will be equal to the number of rooks standing on white triangles. original wordingЄ правильний шестикутник, що розрізаний прямими на 6n^2 правильних трикутників (рис. 2). У них розставлені 2n тур, ніякі дві з яких не б'ють одна одну (тура б'є в напрямках, що паралельні до сторін шестикутника). Доведіть, що якщо розглянути шахове розфарбування всіх 6n^2 правильних трикутників, то тоді кількість тур, що стоять на чорних трикутниках, буде рівна кількості тур, що стоять на білих трикутниках.

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Polynomial $p(x)$ with real coefficients, which is different from the constant, has the following property: for any naturals $n$ and $k$ the $\frac{p(n+1)p(n+2)...p(n+k)}{p(1)p(2)...p(k)}$ is an integer. Prove that this polynomial is divisible by $x$.

Given an acute triangle $ABC$ . It's altitudes $AA_1 , BB_1$ and $CC_1$ intersect at a point $H$ , the orthocenter of $\vartriangle ABC$. Let the lines $B_1C_1$ and $AA_1$ intersect at a point $K$, point $M$ be the midpoint of the segment $AH$. Prove that the circumscribed circle of $\vartriangle MKB_1$ touches the circumscribed circle of $\vartriangle ABC$ if and only if $BA1 = 3A1C$. (Bondarenko Mykhailo)