NL-NWO 2009

In this problem, we consider integers consisting of $5$ digits, of which the rst and last one are nonzero. We say that such an integer is a palindromic product if it satises the following two conditions:

• the integer is a palindrome, (i.e. it doesn't matter if you read it from left to right, or the other way around);
• the integer is a product of two positive integers, of which the first, when read from left to right, is equal to the second, when read from right to left, like $4831$ and $1384$ . For example, $20502$ is a palindromic product, since $102 \cdot 201 = 20502$ , and $20502$ itself is a palindrome. Determine all palindromic products of $5$ digits.

Consider the sequence of integers $0, 1, 2, 4, 6, 9, 12,...$ obtained by starting with zero, adding $1$ , then adding $1$ again, then adding $2$ , and adding $2$ again, then adding $3$ , and adding $3$ again, and so on. If we call the subsequent terms of this sequence $a_0, a_1, a_2, ...$ , then we have $a_0 = 0$ , and $a_{2n-1} = a_{2n-2} + n$ , $a_{2n} = a_{2n-1} + n$ for all integers $n \ge 1$ . Find all integers $k \ge 0$ for which $a_k$ is the square of an integer.

A tennis tournament has at least three participants. Every participant plays exactly one match against every other participant. Moreover, every participant wins at least one of the matches he plays. (Draws do not occur in tennis matches.) Show that there are three participants $A, B$ and $C$ for which the following holds: $A$ wins against $B, B$ wins against $C$ , and $C$ wins against $A$ .

Let $ABC$ be an arbitrary triangle. On the perpendicular bisector of $AB$ , there is a point $P$ inside of triangle $ABC$ . On the sides $BC$ and $CA$ , triangles $BQC$ and $CRA$ are placed externally. These triangles satisfy $\vartriangle BPA \sim \vartriangle BQC \sim \vartriangle CRA$ . (So $Q$ and $A$ lie on opposite sides of $BC$ , and $R$ and $B$ lie on opposite sides of $AC$ .) Show that the points $P, Q, C$ and $R$ form a parallelogram.

We number a hundred blank cards on both sides with the numbers $1$ to $100$ . The cards are then stacked in order, with the card with the number $1$ on top. The order of the cards is changed step by step as follows: at the $1$ st step the top card is turned around, and is put back on top of the stack (nothing changes, of course), at the $2$ nd step the topmost $2$ cards are turned around, and put back on top of the stack, up to the $100$ th step, in which the entire stack of $100$ cards is turned around. At the $101$ st step, again only the top card is turned around, at the $102$ nd step, the top most $2$ cards are turned around, and so on. Show that after a finite number of steps, the cards return to their original positions.