In a school there are 40 different clubs, each of them contains exactly 30 children. For every $i$ from $1$ to $30$ define $n_{i}$ as a number of children who attend exactly $i$ clubs. Prove that it is possible to organize 40 new clubs with 30 children in each of them such that the analogical numbers $n_{1}, n_{2}, \ldots, n_{30}$ will be the same for them.
Let Pascal triangle be an equilateral triangular array of numbers, consisting of $2019$ rows and except for the numbers in the bottom row, each number is equal to the sum of two numbers immediately below it. How many ways to assign each of numbers $a_{0}, a_{1}, \ldots, a_{2018}$ (from left to right) in the bottom row by $0$ or $1$ such that the number $S$ on the top is divisible by $1009$.
Let pairwise different positive integers $a$, $b$, $c$ with $\operatorname{gcd}(a, b, c) = 1$ be such that
$$
a \mid (b - c)^2, \quad b \mid (c - a)^2, \quad c \mid (a - b)^2.
$$
Prove that there is no non-degenerate triangle with side lengths $a$, $b$ and $c$.
Let be given a positive integer $n \gt 1$. Find all polynomials $P(x)$ non constant, with real coefficients such that
$$
P(x) P\left(x^{2}\right) \ldots P\left(x^{n}\right) = P\left(x^{\frac{n(n+1)}{2}}\right)
$$
for all $x \in \mathbb{R}$.
Let $ABC$ be an acute, non-isosceles triangle with $O$, $H$ as circumcenter and orthocenter, respectively. Prove that the nine-point circles of $AHO$, $BHO$, $CHO$ have two common points.
Let $P(x)$ be a monic polynomial of degree $100$ with $100$ distinct noninteger real roots. Suppose that each of polynomials $P\left(2x^{2}-4x\right)$ and $P\left(4x-2x^{2}\right)$ has exactly $130$ distinct real roots. Prove that there exist non constant polynomials $A(x), B(x)$ such that $A(x)B(x)=P(x)$ and $A(x)=B(x)$ has no root in $(-1 ; 1)$.
Let $ABC$ be a triangle, the circle having $BC$ as diameter cuts $AB$, $AC$ at $F$, $E$ respectively. Let $P$ be a point on this circle. Let $C_{0}$, $B_{0}$ be the projections of $P$ upon the sides $AB$, $AC$ respectively. Let $H$ be the orthocenter of the triangle $AB_{0}C_{0}$. Show that $\angle EHF = 90^{\circ}$.
All of the numbers $1, 2, 3, \ldots, 1000000$ are initially colored black. On each move it is possible to choose the number $x$ (among the colored numbers) and change the color of $x$ and of all of the numbers that are not co-prime with $x$ (black into white, white into black). Is it possible to color all of the numbers white?
Find the smallest integer $m$ for which there are positive integers $n \gt k \gt 1$ satisfying the equation
$$
\underbrace{11 \ldots 1}_{n} = \underbrace{11 \ldots 1}_{k} \cdot m.
$$
A chess horse attacks fields at distance $\sqrt{5}$. Let several horses be put on the board $12 \times 12$ such that every square of size $2 \times 2$ contains at least one horse. Find the maximal possible number of cells that are not under attack (a horse doesn't attack its own cell).
How many integers $n$ satisfy the following conditions?
i) $219 \leq n \leq 2019$,
ii) there exist $x, y \in \mathbb{Z}$ such that $1 \leq x \lt n \lt y$ and $y$ is divisible by all integers from $1$ to $n$, except two numbers $x$ and $x+1$.
Let $AD$ be the altitude of the right angled triangle $ABC$ with $\angle A = 90^{\circ}$. Let $DE$ be the altitude of the triangle $ADB$ and $DZ$ be the altitude of the triangle $ADC$ respectively. Let $N$ be chosen on the line $AB$ such that $CN$ is parallel to $EZ$. Let $A'$ be the symmetric of $A$ with respect to the line $EZ$ and $I, K$ the projections of $A'$ onto $AB$ and $AC$ respectively. Prove that $\angle NA'T = \angle ADT$, where $T$ is the intersection point of $IK$ and $DE$.
In square $ABCD$ with side $1$, point $E$ lies on $BC$ and $F$ lies on $CD$ such that $\angle EAB = 20^{\circ}$, $\angle EAF = 45^{\circ}$. Find the length of altitude $AH$ of $\triangle AEF$.
An $11 \times 11$ square is partitioned into 121 smaller $1 \times 1$ squares, 4 of which are painted black, the rest being white. We cut a fully white rectangle (possibly a square) out of the big $11 \times 11$ square. What is the maximal area of the rectangle we can obtain regardless of the positions of the black squares? It is allowed to cut the rectangle along the grid lines.
Determine the maximal number of disjoint crosses (5 squares) which can be put inside an $8 \times 8$ chessboard such that the sides of a cross are parallel to the sides of the chessboard.
Let $ABC$ be an acute, non-isosceles triangle. Take two points $D, E$ inside this triangle such that
$$
\begin{aligned}
& \angle DAB = \angle DCB, \quad \angle DAC = \angle DBC; \\
& \angle EAB = \angle EBC, \quad \angle EAC = \angle ECB.
\end{aligned}
$$
Prove that triangle $ADE$ is right.
Let $n$ be a positive integer and let $a_{1}, a_{2}, \ldots, a_{n}$ be any real numbers. Prove that there exist $m, k \in \{1,2, \ldots, n\}$ such that
$$
\left|\sum_{i=1}^{m} a_{i}-\sum_{i=m+1}^{n} a_{i}\right| \leq\left|a_{k}\right| .
$$
