EGMO 2017 Problem 4

Let $n \geq 1$ be an integer and let $t_{1}<t_{2}<\ldots<t_{n}$ be positive integers. In a group of $t_{n}+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following conditions to hold at the same time: i) The number of games played by each person is one of $t_{1}, t_{2}, \ldots, t_{n}$, ii) For every $i$ with $1 \leq i \leq n$, there is someone who has played exactly $t_{i}$ games of chess.

Gerhard Wöginger, Luxembourg