AUSTRIAN-POLISH 2003

Find all real polynomials $p(x)$ such that $p(x-1)p(x+1)= p(x^2-1)$ .

The sequence $a_0, a_1, a_2, ..$ is defined by $a_0 = a, a_{n+1} = a_n + L(a_n)$ , where $L(m)$ is the last digit of $m$ (eg $L(14) = 4$ ). Suppose that the sequence is strictly increasing. Show that infinitely many terms must be divisible by $d = 3$ . For what other d is this true?

A positive integer $m$ is alpine if $m$ divides $2^{2n+1} + 1$ for some positive integer $n$ . Show that the product of two alpine numbers is alpine.

A triangle with sides a, b, c has area S.

The distances of its centroid from the vertices are x, y, z.

Show that:

if (x + y + z)^2 ≤ (a^2 + b^2 + c^2)/2 + 2S√3, then the triangle is equilateral.

Put $f(n) = \frac{n^n - 1}{n - 1}$ . Show that $n!^{f(n)}$ divides $(n^n)!$ . Find as many positive integers as possible for which $n!^{f(n)+1}$ does not divide $(n^n)!$ .

Given reals $x_1 \ge x_2 \ge ... \ge x_{2003} \ge 0$ , show that $$x_1^n - x_2^n + x_2^n - ... - x_{2002}^n + x_{2003}^n \ge (x_1 - x_2 + x_3 - x_4 + ... - x_{2002} + x_{2003})^n$$ for any positive integer $n$ .

Take any 26 distinct numbers from {1, 2, ... , 100}. Show that there must be a non-empty subset of the $26$ whose product is a square.

What is the smallest number of $5\times 1$ tiles which must be placed on a $31\times 5$ rectangle (each covering exactly $5$ unit squares) so that no further tiles can be placed? How many different ways are there of placing the minimal number (so that further tiles are blocked)? What are the answers for a $52\times 5$ rectangle?