On a $ 50 \times 50$ board, the centers of several unit squares are colored black. Find the maximum number of centers that can be colored black in such a way that no three black points form a right-angled triangle.

Let $ a_1, a_2, ..., a_{300}$ be nonnegative real numbers, with $ \sum_{i=1}^{300} a_i = 1$. Find the maximum possible value of $ \sum_{i \neq j, i|j} a_ia_j$.

Let $ ABC$ be a triangle, $ B_1$ the midpoint of side $ AB$ and $ C_1$ the midpoint of side $ AC$. Let $ P$ be the point of intersection ($ \neq A$) of the circumcircles of triangles $ ABC_1$ and $ AB_1C$. Let $ Q$ be the point of intersection ($ \neq A$) of the line $ AP$ and the circumcircle of triangle $ AB_1C_1$. Prove that $ \frac{AP}{AQ} = \frac{3}{2}$.

Find all positive integers $ n$ such that $ 20^n - 13^n - 7^n$ is divisible by $ 309$.

There are several contestants at a math olympiad. We say that two contestants $ A$ and $ B$ are indirect friends if there are contestants $ C_1, C_2, ..., C_n$ such that $ A$ and $ C_1$ are friends, $ C_1$ and $ C_2$ are friends, $ C_2$ and $ C_3$ are friends, ..., $ C_n$ and $ B$ are friends. In particular, if $ A$ and $ B$ are friends themselves, they are indirect friends as well. Some of the contestants were friends before the olympiad. During the olympiad, some contestants make new friends, so that after the olympiad every contestant has at least one friend among the other contestants. We say that a contestant is special if, after the olympiad, he has exactly twice as indirect friends as he had before the olympiad. Prove that the number of special contestants is less or equal than $ \frac{2}{3}$ of the total number of contestants.

Let $ n \geq 3$ be an odd integer. We denote by $ [-n,n]$ the set of all integers greater or equal than $ -n$ and less or equal than $ n$. Player $ A$ chooses an arbitrary positive integer $ k$, then player $ B$ picks a subset of $ k$ (distinct) elements from $ [-n,n]$. Let this subset be $ S$. If all numbers in $ [-n,n]$ can be written as the sum of exactly $ n$ distinct elements of $ S$, then player $ A$ wins the game. If not, $ B$ wins. Find the least value of $ k$ such that player $ A$ can always win the game.