# 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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# Solution: 2021-21 Different unions

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 $$F$$.

For the new version of POW 2021-21, the best solution was submitted by 이재욱 (전기및전자공학부 2018학번, +4). Congratulations!

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# 2021-23 A regular polygon inscribed in a regular polygon.

Let $$n, m$$ be positive integers where $$m$$ divides $$n$$. When there exists a regular $$n$$-gon with area 1, what is the area of the largest regular $$m$$-gon inscribed in the $$n$$-gon in terms of $$n$$ and $$m$$?

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# Notice on 2021-21

The problem on 2021-21 was written in an ambiguous way, which led the contestants to misunderstand the problem. The problem is updated to be more clear, and anyone is again welcome to submit a solution for the problem.

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# Solution: 2021-22 Sum of fractions

Determine all rational numbers that can be written as
$\frac{1}{n_1} + \frac{1}{n_1 n_2} + \frac{1}{n_1 n_2 n_3} + \dots + \frac{1}{n_1 n_2 n_3 \dots n_k} ,$
where $$n_1, n_2, n_3 \dots, n_k$$ are positive integers greater than $$1$$.

The best solution was submitted by 조정휘 (수리과학과 대학원생, +4). Congratulations!

Other solutions were submitted by 신주홍 (수리과학과 2020학번, +3), 이재욱 (전기및전자공학부 2018학번, +3), 이호빈 (수리과학과 대학원생, +3), 전해구 (기계공학과 졸업생, +3).

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# 2021-22 Sum of fractions

Determine all rational numbers that can be written as
$\frac{1}{n_1} + \frac{1}{n_1 n_2} + \frac{1}{n_1 n_2 n_3} + \dots + \frac{1}{n_1 n_2 n_3 \dots n_k} ,$
where $$n_1, n_2, n_3 \dots, n_k$$ are positive integers greater than $$1$$.

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# Solution: 2021-21 Different unions

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. Determine the maximum possible size of $$F$$.

The best solution was submitted by 전해구 (기계공학과 졸업생, +4). Congratulations!

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# 2021-21 Different unions

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 $$F$$.

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Say a natural number $$n$$ is a cyclically perfect if one can arrange the numbers from 1 to $$n$$ on the circle without a repeat so that the sum of any two consecutive numbers is a perfect square. Show that 32 is the smallest cyclically perfect number. Find the second smallest cyclically perfect number.