CZ-SK-MO 1996

A sequence $(G_n)_{n=0}^{\infty}$ satisfies $G(0) = 0$ and $G(n) = n-G(G(n-1))$ for each $n \in N$ . Show that (a) $G(k) \ge G(k -1)$ for every $k \in N$ ; (b) there is no integer $k$ for which $G(k -1) = G(k) = G(k +1)$ .

Let $AP,BQ$ and $CR$ be altitudes of an acute-angled triangle $ABC$ . Show that for any point $X$ inside the triangle $PQR$ there exists a tetrahedron $ABCD$ such that $X$ is the point on the face $ABC$ at the greatest distance from $D$ (measured along the surface of the tetrahedron).

Given six three-element subsets of a finite set $X$ , show that it is possible to color the elements of $X$ in two colors so that none of the given subsets is in one color

Points $A$ and $B$ on the rays $CX$ and $CY$ respectively of an acute angle $XCY$ are given so that $CX < CA = CB < CY$ . Construct a line meeting the ray $CX$ and the segments $AB,BC$ at $K,L,M$ , respectively, such that $KA \cdot YB = XA \cdot MB = LA\cdot LB \ne 0$ .

For which integers $k$ does there exist a function $f : N \to Z$ such that $f(1995) =1996$ and $f(xy) = f(x)+ f(y)+k f(gcd(x,y))$ for all $x,y \in N$ ?

Let $K,L,M$ be points on sides $AB,BC,CA$ , respectively, of a triangle $ABC$ such that $AK/AB = BL/BC = CM/CA = 1/3$ . Show that if the circumcircles of the triangles $AKM, BLK, CML$ are equal, then so are the incircles of these triangles.