Let $a_1,a_2,\ldots,a_{2n}$ be positive real numbers such that $a_ja_{n+j}=1$ for the values $j=1,2,\ldots,n$. a. Prove that either the average of the numbers $a_1,a_2,\ldots,a_n$ is at least 1 or the average of the numbers $a_{n+1},a_{n+2},\ldots,a_{2n}$ is at least 1. b. Assuming that $n\ge2$, prove that there exist two distinct numbers $j,k$ in the set $\{1,2,\ldots,2n\}$ such that \[|a_j-a_k|<\frac{1}{n-1}.\]

In triangle $ABC$, the bisector of angle $B$ meets the opposite side $AC$ at $B'$. Similarly, the bisector of angle $C$ meets the opposite side $AB$ at $C'$ . Prove that $A=60^{\circ}$ if, and only if, $BC'+CB'=BC$.

There are $n$ people standing on a circular track. We want to perform a number of moves so that we end up with a situation where the distance between every two neighbours is the same. The move that is allowed consists in selecting two people and asking one of them to walk a distance $d$ on the circular track clockwise, and asking the other to walk the same distance on the track anticlockwise. The two people selected and the quantity $d$ can vary from move to move. Prove that it is possible to reach the desired situation (where the distance between every two neighbours is the same) after at most $n-1$ moves.

Let $m,n$ be integers. It is known that there are integers $a,b$ such that $am+bn=1$ if, and only if, the greatest common divisor of $m,n$ is 1. You are not required to prove this. Now suppose that $p,q$ are different odd primes. In each case determine if there are integers $a,b$ such that $ap+bq=1$ so that the given condition is satisfied: a. $p$ divides $b$ and $q$ divides $a$; b. $p$ divides $a$ and $q$ divides $b$; c. $p$ does not divide $a$ and $q$ does not divide $b$.