AUSTRIAN-POLISH 1992

For a natural number $n$ , denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.

Each point on the boundary of a square has to be colored in one color. Consider all right triangles with the vertices on the boundary of the square. Determine the least number of colors for which there is a coloring such that no such triangle has all its vertices of the same color.

For all positive numbers $a, b, c$ prove the inequality $2\sqrt{bc + ca + ab} \le \sqrt{3} \sqrt[3]{(b + c)(c + a)(a + b)}$ .

Let $k$ be a positive integer and $u, v$ be real numbers. Consider $P(x) = (x - u^k) (x - uv) (x -v^k) = x^3 + ax^2 + bx + c$ . (a) For $k = 2$ prove that if $a, b, c$ are rational then so is $uv$ . (b) Is that also true for $k = 3$ ?

Given a circle $k$ with center $M$ and radius $r$ , let $AB$ be a fixed diameter of $k$ and let $K$ be a fixed point on the segment $AM$ . Denote by $t$ the tangent of $k$ at $A$ . For any chord $CD$ through $K$ other than $AB$ , denote by $P$ and Q the intersection points of $BC$ and $BD$ with $t$ , respectively. Prove that $AP\cdot AQ$ does not depend on $CD$ .

A function $f: Z \to Z$ has the following properties: $f (92 + x) = f (92 - x)$ $f (19 \cdot 92 + x) = f (19 \cdot 92 - x)$ ( $19 \cdot 92 = 1748$ ) $f (1992 + x) = f (1992 - x)$ for all integers $x$ . Can all positive divisors of $92$ occur as values of f?

Consider triangles $ABC$ in space. (a) What condition must the angles $\angle A, \angle B , \angle C$ of $\triangle ABC$ fulfill in order that there is a point $P$ in space such that $\angle APB, \angle BPC, \angle CPA$ are right angles? (b) Let $d$ be the longest of the edges $PA,PB,PC$ and let $h$ be the longest altitude of $\triangle ABC$ . Show that $\frac{1}{3}\sqrt6 h \le d \le h$ .

Let $n\ge 3$ be a given integer. Nonzero real numbers $a_1,..., a_n$ satisfy: $\frac{-a_1-a_2+a_3+...a_n}{a_1}=\frac{a_1-a_2-a_3+a_4+...a_n}{a_2}=...=\frac{a_1+...+a_{n-2}-a_{n-1}-a_n}{a_{n-1}}=\frac{-a_1+a_2+...+a_{n-1}-a_n}{a_{n}}$ What values can be taken by the product $\frac{a_2+a_3+...a_n}{a_1}\cdot \frac{a_1+a_3+a_4+...a_n}{a_2}\cdot ...\cdot \frac{+a_1+a_2+...+a_{n-1}}{a_{n}}$ ?

Given an integer $n > 1$ , consider words composed of $n$ letters $A$ and $n$ letters $B$ . A word $X_1...X_{2n}$ is said to belong to set $R(n)$ (respectively, $S(n)$ ) if no initial segment (respectively, exactly one initial segment) $X_1...X_k$ with $1 \le k < 2n$ consists of equally many letters $A$ and $B$ . If $r(n)$ and $s(n)$ denote the cardinalities of $R(n)$ and $S(n)$ respectively, compute $s(n)/r(n)$ .