Ana and Beto are playing a game. Ana writes a whole number on the board. Beto then has the right to erase the number and add $2$ to it, or erase the number and subtract $3$, as many times as he wants. Beto wins if he can get $2021$ after a finite number of stages; otherwise, Ana wins. Which player has a winning strategy?

Let $ABC$ be a right triangle with right angle at $ B$ and $\angle C=30^o$. If $M$ is midpoint of the hypotenuse and $I$ the incenter of the triangle, show that $ \angle IMB=15^o$.

Coins are placed in some squares on a $n\times n$ board. Each coin can be moved towards the square symmetrical with respect to either of the two diagonals, as long as that square is empty. The initial coin setup is said to be good , if any coin can make the first move. (a) Determine the maximum number of coins $M$ that can be placed on the $n\times n$ board, such that the configuration is good. (b) Calculate the total number of good configurations that have exactly $M$ coins.

How many numbers $\overline{abcd}$ with different digits satisfy the following property: if we replace the largest digit with the digit $1$ results in a multiple of $30$?

Circle $o$ contains the circles $m$ , $p$ and $r$, such that they are tangent to $o$ internally and any two of them are tangent between themselves. The radii of the circles $m$ and $p$ are equal to $x$ . The circle $r$ has radius $1$ and passes through the center of the circle $o$. Find the value of $x$ .

Two positive integers $n,m\ge 2$ are called allies if when written as a product of primes (not necessarily different): $n=p_1p_2...p_s$ and $m=q_1q_2...q_t$, turns out that: $$p_1 + p_2 + ... + p_s = q_1 + q_2 + ... + q_t$$(a) Show that the biggest ally of any positive integer has to have only $2$ and $3$ in its prime factorization. (b) Find the biggest number which is allied of $2021$ .