PAMO 2010

a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers. b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?

How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?

In an acute-angled triangle $ABC$ , $CF$ is an altitude, with $F$ on $AB$ , and $BM$ is a median, with $M$ on $CA$ . Given that $BM=CF$ and $\angle MBC=\angle FCA$ , prove that triangle $ABC$ is equilateral.

Seven distinct points are marked on a circle of circumference $c$ . Three of the points form an equilateral triangle and the other four form a square. Prove that at least one of the seven arcs into which the seven points divide the circle has length less than or equal $\frac{c}{24}$ .

A sequence $a_0,a_1,a_2,\ldots ,a_n,\ldots$ of positive integers is constructed as follows:

• if the last digit of $a_n$ is less than or equal to $5$ then this digit is deleted and $a_{n+1}$ is the number consisting of the remaining digits. (If $a_{n+1}$ contains no digits the process stops.)
• otherwise $a_{n+1}=9a_n$ .

Can one choose $a_0$ so that an infinite sequence is obtained?

Does there exist a function $f:\mathbb{Z}\to\mathbb{Z}$ such that $f(x+f(y))=f(x)-y$ for all integers $x$ and $y$ ?