MathDB

Let

n\geq1

be an integer and let

t_1<t_2<\dots<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 two conditions to hold at the same time:
(i) The number of games played by each person is one of

t_1,t_2,\dots,t_n

.
(ii) For every

i

with

1\leq i\leq n

, there is someone who has played exactly

t_i

games of chess.

Problem Statement