CZ-SK-MO 1992

For a permutation $p(a_1,a_2,...,a_{17})$ of $1,2,...,17$ , let $k_p$ denote the largest $k$ for which $a_1 +...+a_k < a_{k+1} +...+a_{17}$ . Find the maximum and minimum values of $k_p$ and find the sum $\sum_{p} k_p$ over all permutations $p$ .

Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$

Let $S(n)$ denote the sum of digits of $n \in N$ . Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$

Solve the equation $\cos 12x = 5\sin 3x+9\ tan ^2x+\ cot ^2x$

The function $f : (0,1) \to R$ is defined by $f(x) = x$ if $x$ is irrational, $f(x) = \frac{p+1}{q}$ if $x =\frac{p}{q}$ , where $(p,q) = 1$ . Find the maximum value of $f$ on the interval $(7/8,8/9)$ .

Let $ABC$ be an acute triangle. The altitude from $B$ meets the circle with diameter $AC$ at points $P,Q$ , and the altitude from $C$ meets the circle with diameter $AB$ at $M,N$ . Prove that the points $M,N,P,Q$ lie on a circle.