KURSCHAK 1953

$A$ and $B$ are any two subsets of $\{1, 2,...,n - 1\}$ such that $|A| +|B|> n - 1$ . Prove that one can find $a$ in $A$ and $b$ in $B$ such that $a + b = n$ .

$n$ and $d$ are positive integers such that $d$ divides $2n^2$ . Prove that $n^2 + d$ cannot be a square.

$ABCDEF$ is a convex hexagon with all its sides equal. Also $\angle A + \angle C + \angle E = \angle B + \angle D + \angle F$ . Show that $\angle A = \angle D$ , $\angle B = \angle E$ and $\angle C = \angle F$ .