TR-MO 1995

Let $m_{1},m_{2},\ldots,m_{k}$ be integers with $2\leq m_{1}$ and $2m_{1}\leq m_{i+1}$ for all $i$ . Show that for any integers $a_{1},a_{2},\ldots,a_{k}$ there are infinitely many integers $x$ which do not satisfy any of the congruences $$x\equiv a_{i}\ (\bmod \ m_{i}),\ i=1,2,\ldots k.$$

Let $ABC$ be an acute triangle and let $k_{1},k_{2},k_{3}$ be the circles with diameters $BC,CA,AB$ , respectively. Let $K$ be the radical center of these circles. Segments $AK,CK,BK$ meet $k_{1},k_{2},k_{3}$ again at $D,E,F$ , respectively. If the areas of triangles $ABC,DBC,ECA,FAB$ are $u,x,y,z$ , respectively, prove that $$u^{2}=x^{2}+y^{2}+z^{2}.$$

In a triangle $ABC$ with $AB\neq AC$ , the internal and external bisectors of angle $A$ meet the line $BC$ at $D$ and $E$ respectively. If the feet of the perpendiculars from a point $F$ on the circle with diameter $DE$ to $BC,CA,AB$ are $K,L,M$ , respectively, show that $KL=KM$ .

Let $t(A)$ denote the sum of elements of a nonempty set $A$ of integers, and define $t(\emptyset)=0$ . Find a set $X$ of positive integers such that for every integers $k$ there is a unique ordered pair of disjoint subsets $(A_{k},B_{k})$ of $X$ such that $t(A_{k})-t(B_{k}) = k$ .