Find the minimal positive integer $n$ such that no matter what $n$ distinct numbers from $1$ to $1000$ you choose, such that no two are divisible by a square of the same prime, one of the chosen numbers is a square of prime. D. Zmiaikou

An acute-angled triangle $ABC$ with an altitude $AD$ and orthocenter $H$ are given. $AD$ intersects the circumcircle of $ABC$ $\omega$ at $P$. $K$ is a point on segment $BC$ such that $KC=BD$. The circumcircle of $KPH$ intersects $\omega$ at $Q$ and $BC$ at $N$. A line perpendicular to $PQ$ and passing through $N$ intersects $AD$ at $T$. Prove that the center of $\omega$ lies on line $TK$. U. Maksimenkau

Prove that for any real numbers $a,b,c,d \geq \frac{1}{3}$ the following inequality holds: $$\sqrt{\frac{a^6}{b^4+c^3}+\frac{b^6}{c^4+d^3}+\frac{c^6}{d^4+a^3}+\frac{d^6}{a^4+b^3}}\geq \frac{a+b+c+d}{4}$$D. Zmiaikou

Two permutations of $1,\ldots, n$ are written on the board: $a_1,\ldots,a_n$ $b_1,\ldots,b_n$ A move consists of one of the following two operations: 1) Change the first row to $b_{a_1},\ldots,b_{a_n}$ 2) Change the second row to $a_{b_1},\ldots,a_{b_n}$ The starting position is: $2134\ldots n$ $234\ldots n1$ Is it possible by finitely many moves to get: $2314\ldots n$ $234 \ldots n1$? D. Zmiaikou

A sequence $\{y_i\}$ is given, where $y_0=-\frac{1}{4},y_1=0$. For every positive integer $n$ the following equality holds: $$y_{n-1}+y_{n+1}=4y_n+1$$Prove that for every positive integer $n$ the number $2y_{2n}+\frac{3}{2}$ a) is a positive integer b) is a square of a positive integer D. Zmiaikou

A positive integer $n$ is given. Consider all polynomials $P(x)=x^n+a_{n-1}x^{n-1}+\ldots+a_0$, whose coefficients are nonnegative integers, not exceeding $100$. Call $P$ reducible if it can be factored into two non-constant polynomials with nonnegative integer coeffiecients, and irreducible otherwise. Prove that the number of irreducible polynomials is at least twice as big as the number of reducible polynomials. D. Zmiaikou

A right triangle $ABC$ ($\angle A=90$) is inscribed in a circle $\omega$. Tangent to $\omega$ at $A$ intersects $BC$ at $P$, $B$ lies between $P$ and $C$. Let $M$ be the midpoint of the minor arc $AB$. $MP$ intersects $\omega$ at $Q$. Point $X$ lies on a ray $PA$ such that $\angle XCB=90$. Prove that line $XQ$ passes through the orthocenter of the triangle $ABO$ Mayya Golitsyna

There are $k$ cities in Belarus and $k$ cities in Armenia, between some cities there are non-directed flights. From any Belarusian city there are exactly $n$ flights to Armenian cities, and for every pair of Armenian cities exactly two Belarusian cities have flights to both of the Armenian cities. a) Prove that from every Armenian city there are exactly $n$ flights to Belarusian cities. b) Prove that there exists a flight route in which every city is visited at most once and that consists of at least $\lfloor \frac{(n+1)^2}{4} \rfloor$ cities in both countries. D. Gorovoy

Triangles $ABC$ and $DEF$, having a common incircle of radius $R$, intersect at points $X_1, X_2, \ldots , X_6$ and form six triangles (see the figure below). Let $r_1, r_2,\ldots, r_6$ be the radii of the inscribed circles of these triangles, and let $R_1, R_2, \ldots , R_6$ be the radii of the inscribed circles of the triangles $AX_1F, FX_2B, BX_3D, DX_4C, CX_5E$ and $EX_6A$ respectively. Prove that \[ \sum_{i=1}^{6} \frac{1}{r_i} < \frac{6}{R}+\sum_{i=1}^{6} \frac{1}{R_i} \]U. Maksimenkau

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any reals $x \neq y$ the following equality is true: $$f(x+y)^2=f(x+y)+f(x)+f(y)$$D. Zmiaikou

Olya and Tolya are playing a game on $[0,1]$ segment. In the beginning it is white. In the first round Tolya chooses a number $0 \leq l \leq 1$, and then Olya chooses a subsegment of $[0,1]$ of length $l$ and recolors every its point to the opposite color(white to black, black to white). In the next round players change roles, etc. The game lasts $2024$ rounds. Let $L$ be the sum of length of white segments after the end of the game. If $L > \frac{1}{2}$ Olya wins, otherwise Tolya wins. Which player has a strategy to guarantee his win? A. Naradzetski

Points $A_1, \ldots A_n$ with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points $B_1, \ldots ,B_n$ with integer coordinates such that $A_iA_j=B_iB_j$ for every pair $1 \leq i \leq j \leq n$ N. Sheshko, D. Zmiaikou

Six integers $a,b,c,d,e,f$ satisfy: $\begin{cases} ace+3ebd-3bcf+3adf=5 \\ bce+acf-ade+3bdf=2 \end{cases}$ Find all possible values of $abcde$ D. Bazyleu

Let $f(x)=x^2+bx+c$, where $b,c \in \mathbb{R}$ and $b>0$ Do there exist disjoint sets $A$ and $B$, whose union is $[0,1]$ and $f(A)=B$, where $f(X)=\{f(x), x \in X\}$ D. Zmiaikou

An isosceles triangle $ABC$ is given($AB=BC$). Point $D$ lies inside of it such that $\angle ADC=150$, $E$ lies on $CD$ such that $AE=AB$. It turned out that $\angle EBC+\angle BAE=60$. Prove that $\angle BDC+\angle CAE=90$ D. Vasilyev

Given positive integers $n$ and $k \leq n$. Consider an equilateral triangular board with side $n$, which consists of circles: in the first (top) row there is one circle, in the second row there are two circles, $\ldots$ , in the bottom row there are $n$ circles (see the figure below). Let us place checkers on this board so that any line parallel to a side of the triangle (there are $3n$ such lines) contains no more than $k$ checkers. Denote by $T(k, n)$ the largest possible number of checkers in such a placement. a) Prove that the following upper bound is true: $$T(k,n) \leq \lfloor \frac{k(2n+1)}{3} \rfloor$$b) Find $T(1,n)$ and $T(2,n)$ D. Zmiaikou