Let $ABC$ be a triangle and $D$ the midpoint of the side $BC$. Define points $E$ and $F$ on $AC$ and $B$, respectively, such that $DE=DF$ and $\angle EDF =\angle BAC$. Prove that $$DE\geq \frac {AB+AC} 4.$$

Initally a pair $(x, y)$ is written on the board, such that exactly one of it's coordinates is odd. On such a pair we perform an operation to get pair $(\frac x 2, y+\frac x 2)$ if $2|x$ and $(x+\frac y 2, \frac y 2)$ if $2|y$. Prove that for every odd $n>1$ there is a even positive integer $b<n$ such that starting from the pair $(n, b)$ we will get the pair $(b, n)$ after finitely many operations.

A function $f:\mathbb{N} \rightarrow \mathbb{N} $ is called nice if $f^a(b)=f(a+b-1)$, where $f^a(b)$ denotes $a$ times applied function $f$. Let $g$ be a nice function, and an integer $A$ exists such that $g(A+2018)=g(A)+1$. a) Prove that $g(n+2017^{2017})=g(n)$ for all $n \geq A+2$. b) If $g(A+1) \neq g(A+1+2017^{2017})$ find $g(n)$ for $n <A$.

We have an $n \times n$ square divided into unit squares. Each side of unit square is called unit segment. Some isoceles right triangles of hypotenuse $2$ are put on the square so all their vertices are also vertices of unit squares. For which $n$ it is possible that every unit segment belongs to exactly one triangle(unit segment belongs to a triangle even if it's on the border of the triangle)?

Let $n \geq 2$ be a positive integer and $\{x_i\}_{i=0}^n$ a sequence such that not all of its elements are zero and there is a positive constant $C_n$ for which: (i) $x_1+ \dots +x_n=0$, and (ii) for each $i$ either $x_i\leq x_{i+1}$ or $x_i\leq x_{i+1} + C_n x_{i+2}$ (all indexes are assumed modulo $n$). Prove that a) $C_n\geq 2$, and b) $C_n=2$ if and only $2 \mid n$.

Let $k$ be a positive integer and let $n$ be the smallest number with exactly $k$ divisors. Given $n$ is a cube, is it possible that $k$ is divisible by a prime factor of the form $3j+2$?