KURSCHAK 1960

Among any four people at a party there is one who has met the three others before the party. Show that among any four people at the party there must be one who has met everyone at the party before the party

Let $a_1 = 1, a_2, a_3,...$ : be a sequence of positive integers such that $$a_k < 1 + a_1 + a_2 +... + a_{k-1}$$ for all $k > 1$ . Prove that every positive integer can be expressed as a sum of $a_i$ s.

$E$ is the midpoint of the side $AB$ of the square $ABCD$ , and $F, G$ are any points on the sides $BC$ , $CD$ such that $EF$ is parallel to $AG$ . Show that $FG$ touches the inscribed circle of the square.