# Solution: 2021-24 The squares of wins and losses

There are $$n$$ people participating to a chess tournament and every two players play one game. There are no draws. Let $$a_i$$ be the number of wins of the $$i$$-th player and $$b_i$$ be the number of losses of the $$i$$-th player. Prove that
$\sum_{i\in [n]} a_i^2 = \sum_{i\in [n]} b_i^2.$

The best solution was submitted by 구재현 (전산학부 2017학번, +4). Congratulations!

Other solutions were submitted by 이도현 (수리과학과 2018학번, +3), 이재욱 (전기및전자공학부 2018학번, +3), 이충명 (기계공학과 대학원생, +3), 이호빈 (수리과학과 대학원생, +3), 전해구 (기계공학과 졸업생, +3). Late solutions were not graded.

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# 2021-24 The squares of wins and losses

There are $$n$$ people participating to a chess tournament and every two players play one game. There are no draws. Let $$a_i$$ be the number of wins of the $$i$$-th player and $$b_i$$ be the number of losses of the $$i$$-th player. Prove that
$\sum_{i\in [n]} a_i^2 = \sum_{i\in [n]} b_i^2.$

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Let $$F$$ be a family of nonempty subsets of $$[n]=\{1,\dots,n\}$$ such that no two disjoint subsets of $$F$$ have the same union. In other words, for $$F =\{ A_1,A_2,\dots, A_k\},$$ there exists no two sets $$I, J\subseteq [k]$$ with $$I\cap J =\emptyset$$ and $$\bigcup_{i\in I}A_i = \bigcup_{j\in J} A_j$$. Determine the maximum possible size of $$F$$.