Determine the natural number $a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$ where $p, q$ and $r$ are prime positive numbers.

We call word a sequence of letters $\overline {l_1l_2...l_n}, n\ge 1$ . A word $\overline {l_1l_2...l_n}, n\ge 1$ is called palindrome if $l_k=l_{n-k+1}$ , for any $k, 1 \le k \le n$. Consider a word $X=\overline {l_1l_2...l_{2014}}$ in which $ l_k\in\{A,B\}$ , for any $k, 1\le k \le 2014$. Prove that there are at least $806$ palindrome words to ''stick" together to get word $X$.

Let $ABC$ be a triangle with $\angle A<90^o, AB \ne AC$. Denote $H$ the orthocenter of triangle $ABC$, $N$ the midpoint of segment $[AH]$, $M$ the midpoint of segment $[BC]$ and $D$ the intersection point of the angle bisector of $\angle BAC$ with the segment $[MN]$. Prove that $<ADH=90^o$

Consider the real numbers $a_1,a_2,...,a_{2n}$ whose sum is equal to $0$. Prove that among pairs $(a_i,a_j) , i<j$ where $ i,j \in \{1,2,...,2n\} $ .there are at least $2n-1$ pairs with the property that $a_i+a_j\ge 0$.

Two circles $\gamma_1$ and $\gamma_2$ cross one another at two points; let $A$ be one of these points. The tangent to $\gamma_1$ at $A$ meets again $\gamma_2$ at $B$, the tangent to $\gamma_2$ at $A$ meets again $\gamma_1$ at $C$, and the line $BC$ meets again $\gamma_1$ and $\gamma_2$ at $D_1$ and $D_2$, respectively. Let $E_1$ and $E_2$ be interior points of the segments $AD_1$ and $AD_2$, respectively, such that $AE_1 = AE_2$. The lines $BE_1$ and $AC$ meet at $M$, the lines $CE_2$ and $AB$ meet at $N$, and the lines $MN$ and $BC$ meet at $P$. Show that the line $PA$ is tangent to the circle $ABC$.

Let $S$ be a set of positive integers such that $\lfloor \sqrt{x}\rfloor =\lfloor \sqrt{y}\rfloor $ for all $x, y \in S$. Show that the products $xy$, where $x, y \in S$, are pairwise distinct.

Given any integer $n \ge 2$, show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.

Let $n$ be a positive integer and let $\triangle$ be the closed triangular domain with vertices at the lattice points $(0, 0), (n, 0)$ and $(0, n)$. Determine the maximal cardinality a set $S$ of lattice points in $\triangle$ may have, if the line through every pair of distinct points in $S$ is parallel to no side of $\triangle$.