Determine the least positive integer $n{}$ for which the following statement is true: the product of any $n{}$ odd consecutive positive integers is divisible by $45$.

In a convex hexagon the value of each angle is $120^{\circ}$. The perimeter of the hexagon equals $2$. Prove that this hexagon can be covered by a triangle with perimeter at most $3$.

The numbers $1, 2, 3,\ldots, 2\underbrace{00\ldots0}_{100 \text{ zeroes}}2$ are written on the board. Is it possible to paint half of them red and remaining ones blue, so that the sum of red numbers is divisible by the sum of blue ones?

Pasha and Vova play the game crossing out the cells of the $3\times 101$ board by turns. At the start, the central cell is crossed out. By one move the player chooses the diagonal (there can be $1, 2$ or $3$ cells in the diagonal) and crosses out cells of this diagonal which are still uncrossed. At least one new cell must be crossed out by any player's move. Pasha begins, the one who can not make any move loses. Who has a winning strategy?

Is it possible to fill a table $1\times n$ with pairwise distinct integers such that for any $k = 1, 2,\ldots, n$ one can find a rectangle $1\times k$ in which the sum of the numbers equals $0$ if a) $n= 11$; b) $n= 12$?

Let $a, b, c$ be positive integers such that $$\gcd(a, b) + \text{lcm}(a, b) = \gcd(a, c) + \text{lcm}(a, c).$$Does it follow from this that $b = c$?

Sasha has $10$ cards with numbers $1, 2, 4, 8,\ldots, 512$. He writes the number $0$ on the board and invites Dima to play a game. Dima tells the integer $0 < p < 10, p$ can vary from round to round. Sasha chooses $p$ cards before which he puts a “$+$” sign, and before the other cards he puts a “$-$" sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha’s moves.

Let $ABC$ be an equilateral triangle with the side length equals $a+ b+ c$. On the side $AB{}$ of the triangle $ABC$ points $C_1$ and $C_2$ are chosen, on the side $BC$ points $A_1$ and $A_2$, arc chosen, and on the side $CA$ points $B_1$ and $B_2$ are chosen such that $A_1A_2 = CB_1 = BC_2 = a, B_1B_2 = AC_1 = CA_2 = b, C_1C_2 = BA_1 = AB_2 = c$. Let the point $A^{’}$ be such that the triangle $A^{'} B_2C_1$ is equilateral, and the points $A$ and $A^{'}$ lie on different sides of the line $B_2C_1$. Similarly, the points $B^{’}$ and $C^{'}$ are constructed (the triangle $B^{'} C_2A_1$ is equilateral, and the points $B$ and $B^{’}$ lie on different sides of the line $C_2A_1$; the triangle $C^{'} A_2B_1$ is equilateral, and the points $C$ and $C^{'}$ lie on different sides of the line $A_2B_1$). Prove that the triangle $A^{'}B^{'}C^{'}$ is equilateral.

Let $n{}$ and $m$ be positive integers, $n>m>1$. Let $n{}$ divided by $m$ have partial quotient $q$ and remainder $r$ (so that $n = qm + r$, where $r\in\{0,1,...,m-1\}$). Let $n-1$ divided by $m$ have partial quotient $q^{'}$ and remainder $r^{'}$. a) It appears that $q+q^{'} =r +r^{'} = 99$. Find all possible values of $n{}$. b) Prove that if $q+q^{'} =r +r^{'}$, then $2n$ is a perfect square.

Given are reals $a, b$. Prove that at least one of the equations $x^4-2b^3x+a^4=0$ and $x^4-2a^3x+b^4=0$ has a real root. Proposed by N. Agakhanov

a) Determine if there exists a convex hexagon $ABCDEF$ with $$\angle ABD + \angle AED > 180^{\circ},$$$$\angle BCE + \angle BFE > 180^{\circ},$$$$\angle CDF + \angle CAF > 180^{\circ}.$$b) The same question, with additional condition, that diagonals $AD, BE,$ and $CF$ are concurrent.

Let $n>k>1$ be positive integers and let $G$ be a graph with $n$ vertices such that among any $k$ vertices, there is a vertex connected to the rest $k-1$ vertices. Find the minimal possible number of edges of $G$. Proposed by V. Dolnikov

Let $n \leq 100$ be an integer. Hare puts real numbers in the cells of a $100 \times 100$ table. By asking Hare one question, Wolf can find out the sum of all numbers of a square $n \times n$, or the sum of all numbers of a rectangle $1 \times (n - 1)$ (or $(n - 1) \times 1$). Find the greatest $n{}$ such that, after several questions, Wolf can find the numbers in all cells, with guarantee.

Numbers $1, 2,\ldots, n$ are written on the board. By one move, we replace some two numbers $ a, b$ with the number $a^2-b{}$. Find all $n{}$ such that after $n-1$ moves it is possible to obtain $0$.

Let $ABC$ be an acute-angled triangle, and let $AA_1, BB_1, CC_1$ be its altitudes. Points $A', B', C'$ are chosen on the segments $AA_1, BB_1, CC_1$, respectively, so that $\angle BA'C = \angle AC'B = \angle CB'A = 90^{o}$. Let segments $AC'$ and $CA'$ intersect at $B"$; points $A", C"$ are defined similarly. Prove that hexagon $A'B"C'A"B'C"$ is circumscribed.