TW-TST 1998

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 $[n+3-\frac{18m}{n}]$ .