AUSTRIAN-POLISH 1982

Find all pairs $(n, m)$ of positive integers such that $gcd ((n + 1)^m - n, (n + 1)^{m+3} - n) > 1$ .

Let $F$ be a closed convex region inside a circle $C$ with center $O$ and radius $1$ . Furthermore, assume that from each point of $C$ one can draw two rays tangent to $F$ which form an angle of $60^o$ . Prove that $F$ is the disc centered at $O$ with radius $1/2$ .

If $n \ge 2$ is an integer, prove the equality $$\prod_{k=1}^n \tan \frac{\pi}{3}\left(1+\frac{3^k}{3^n-1}\right)=\prod_{k=1}^n \cot \frac{\pi}{3}\left(1-\frac{3^k}{3^n-1}\right)$$

Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$ . For any positive integer $x_1$ , define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$ . Prove or disprove that the sequence $(x_n)$ is necessarily bounded.

Show that [0,1] cannot be partitioned into two disjoints sets A and B such that B=A+a for some real a.

An integer $a$ is given. Find all real-valued functions $f (x)$ defined on integers $x \ge a$ , satisfying the equation $f (x+y) = f (x) f (y)$ for all $x,y \ge a$ with $x + y \ge a$ .

Find the triple of positive integers $(x,y,z)$ with $z$ least possible for which there are positive integers $a, b, c, d$ with the following properties: (i) $x^y = a^b = c^d$ and $x > a > c$ (ii) $z = ab = cd$ (iii) $x + y = a + b$ .

Let $P$ be a point inside a regular tetrahedron ABCD with edge length $1$ . Show that $$d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2$$ , with equality only when $P$ is the centroid of $ABCD$ . Here $d(P,XY)$ denotes the distance from point $P$ to line $XY$ .

Define $S_n=\sum_{j,k=1}^{n} \frac{1}{\sqrt{j^2+k^2}}$ . Find a positive constant $C$ such that the inequality $n\le S_n \le Cn$ holds for all $n \ge 3$ . (Note. The smaller $C$ , the better the solution.)