Let $m,n$ are positive integers. a)Prove that $(m,n)=2\sum_{k=0}^{m-1}[\frac{kn}{m}]+m+n-mn$. b)If $m,n\geq 2$, prove that $\sum_{k=0}^{m-1}[\frac{kn}{m}]=\sum_{k=0}^{n-1}[\frac{km}{n}]$.

Does there exist a solution $(x,y,z,u,v)$ in integers greater than $1998$ to the equation $x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65$?

Let $ m,n$ be positive integers, and let $ F$ be a family of $ m$-element subsets of $ \{1,2,...,n\}$ satisfying $ A\cap B \not = \emptyset$ for all $ A,B\in F$. Determine the maximum possible number of elements in $ F$.

Let $I$ be the incenter of triangle $ABC$. Lines $AI$, $BI$, $CI$ meet the sides of $\triangle ABC$ at $D$, $E$, $F$ respectively. Let $X$, $Y$, $Z$ be arbitrary points on segments $EF$, $FD$, $DE$, respectively. Prove that $d(X, AB) + d(Y, BC) + d(Z, CA) \leq XY + YZ + ZX$, where $d(X, \ell)$ denotes the distance from a point $X$ to a line $\ell$.

For a positive integer $n$, let $\omega(n)$ denote the number of positive prime divisors of $n$. Find the smallest positive tinteger $k$ such that $2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}$.

In a group of $n\geq 4$ persons, every three who know each other have a common signal. Assume that these signals are not repeatad and that there are $m\geq 1$ signals in total. For any set of four persons in which there are three having a common signal, the fourth person has a common signal with at most one of them. Show that there three persons who have a common signal, such that the number of persons having no signal with anyone of them does not exceed $[n+3-\frac{18m}{n}]$.