ZA-SAMO 1998

Find the maximum value of $$\sin{2\alpha} + \sin{2\beta} + \sin{2\gamma}$$ where $\alpha,\beta$ and $\gamma$ are positive and $\alpha + \beta + \gamma = 180^{\circ}$ .

$A,\ B,\ C,\ D,\ E$ and $F$ lie (in that order) on the circumference of a circle. The chords $AD,\ BE$ and $CF$ are concurrent. $P,\ Q$ and $R$ are the midpoints of $AD,\ BE$ and $CF$ respectively. Two further chords $AG \parallel BE$ and $AH \parallel CF$ are drawn. Show that $PQR$ is similar to $DGH$ .

In a group of people, every two people have exactly one friend in common. Prove that there is a person who is a friend of everyone else.

Prove that $$\gcd{\left({n \choose 1},{n \choose 2},\dots,{n \choose {n - 1}}\right)}$$ is a prime if $n$ is a power of a prime, and 1 otherwise.

You are given $n$ squares, not necessarily all of the same size, which have total area 1. Is it always possible to place them without overlapping in a square of area 2?