AUSTRIAN-POLISH 1996

Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties: (i) $n$ has exactly $k$ digits (in decimal representation), (ii) all the digits of $n$ are odd, (iii) $n$ is divisible by $5$ , (iv) the number $m = n/5$ has $k$ odd digits

A convex hexagon $ABCDEF$ satisfies the following conditions:

1. $AB\parallel DE$ , $BC\parallel EF$ , and $CD\parallel FA$ .
2. The distances between these pairs of parallel lines are the same.
3. $\angle FAB = \angle CDE = 90^\circ$ Prove that the diagonals $BE$ and $CF$ of the hexagon intersect with angle $45$ degrees.

$\bullet$ Thank you dear Babis Stergiou for your translation. :P

The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0,P_{1}(x)=x$ and $$P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) n\geq 2$$ For every natural number $n\geq 1$ , find all real numbers $x$ satisfying the equation $P_{n}(x)=0$ .

Real numbers $x,y,z, t$ satisfy $x + y + z +t = 0$ and $x^2+ y^2+ z^2+t^2 = 1$ .

Prove that $- 1 \le xy + yz + zt + tx \le 0$ .

A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Show that there exists a sphere tangent to all the edges of $P$ .

Given natural numbers $n > k > 1$ , find all real solutions $x_1,..., x_n$ of the system $$x_i^3(x_i^2 + x_{i+1}^2+... +x_{i+k-1}^2) = x_{i-1}^2$$ for 1 $\le i \le n$ . Here $x_{n+i} = x_i$ for all $i$ .

Prove there are no such integers $k, m$ which satisfy $k \ge 0, m \ge 0$ and $k!+48=48(k+1)^m$ .

Show that there is no polynomial $P(x)$ of degree $998$ with real coefficients which satisfies $P(x^2 + 1) = P(x)^2 - 1$ for all $x$ .

For any triple $(a, b, c)$ of positive integers, not all equal, We are given sufficiently many rectangular blocks of size $a \times b \times c$ . We use these blocks to fill up a cubic box of edge $10$ . (a) Assume we have used at least $100$ blocks. Show that there are two blocks, one of which is a translate of the other. (b) Find a number smaller than $100$ (the smaller, the better) for which the above statement still holds.