Find all pairs of integers $(m,n)$ such that an $m\times n$ board can be totally covered with $1\times 3$ and $2 \times 5$ pieces.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$ we have \[f(xf(x)+f(y)) = f(x)^2 + y\]

Given the triangle $ABC$ we consider the points $X,Y,Z$ such that the triangles $ABZ,BCX,CAZ$ are equilateral, and they don't have intersection with $ABC$. Let $B'$ be the midpoint of $BC$, $N'$ the midpoint of $CY$, and $M,N$ the midpoints of $AZ,CX$, respectively. Prove that $B'N' \bot MN$.

We have $ 150$ numbers $ x_1,x_2, \cdots , x_{150}$ each of which is either $ \sqrt 2 +1$ or $ \sqrt 2 -1$ We calculate the following sum: $ S=x_1x_2 +x_3x_4+ x_5x_6+ \cdots + x_{149}x_{150}$ Can we choose the $ 150$ numbers such that $ S=121$? And what about $ S=111$?

Let $n,p$ be integers such that $n>1$ and $p$ is a prime. If $n\mid p-1$ and $p\mid n^3-1$, show that $4p-3$ is a perfect square.

We say that a group of $k$ boys is $n-acceptable$ if removing any boy from the group one can always find, in the other $k-1$ group, a group of $n$ boys such that everyone knows each other. For each $n$, find the biggest $k$ such that in any group of $k$ boys that is $n-acceptable$ we must always have a group of $n+1$ boys such that everyone knows each other.