NO-ABEL 1998

Let $a_0,a_1,a_2,...$ be an infinite sequence of positive integers such that $a_0 = 1$ and $a_i^2 > a_{i-1}a_{i+1}$ for all $i > 0$ . (a) Prove that $a_i < a_1^i$ for all $i > 1$ . (b) Prove that $a_i > i$ for all $i$ .

Let be given an $n \times n$ chessboard, $n \in N$ . We wish to tile it using particular tetraminos which can be rotated. For which $n$ is this possible if we use (a) $T$ -tetraminos (b) both kinds of $L$ -tetraminos?

Let $n$ be a positive integer. (a) Prove that $1^5 +3^5 +5^5 +...+(2n-1)^5$ is divisible by $n$ . (b) Prove that $1^3 +3^3 +5^3 +...+(2n-1)^3$ is divisible by $n^2$ .

Let $A,B,P$ be points on a line $\ell$ , with $P$ outside the segment $AB$ . Lines $a$ and $b$ pass through $A$ and $B$ and are perpendicular to $\ell$ . A line $m$ through $P$ , which is neither parallel nor perpendicular to $\ell$ , intersects $a$ and $b$ at $Q$ and $R$ , respectively. The perpendicular from $B$ to $AR$ meets $a$ and $AR$ at $S$ and $U$ , and the perpendicular from $A$ to $BQ$ meets $b$ and $BQ$ at $T$ and $V$ , respectively. (a) Prove that $P,S,T$ are collinear. (b) Prove that $P,U,V$ are collinear.