# Solution: 2023-12 Pairs promoting diversity

Let $$p$$ be a prime number at least three and let $$k$$ be a positive integer smaller than $$p$$. Given $$a_1,\dots, a_k\in \mathbb{F}_p$$ and distinct elements $$b_1,\dots, b_k\in \mathbb{F}_p$$, prove that there exists a permutation $$\sigma$$ of $$[k]$$ such that the values of $$a_i + b_{\sigma(i)}$$ are distinct modulo $$p$$.

The best solution was submitted by 이명규 (KAIST 전산학부, +4). Congratulations!

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 최민규 (한양대학교 의학대학 졸업, +3), Anar Rzayev (KAIST 전산학부 19학번, +3). Late solutions were not graded.

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# Solution: 2023-11 Possible outcomes of sums

Let $$S$$ be a set of distinct $$20$$ integers. A set $$T_A$$ is defined as $$T_A:=\{ s_1+s_2+s_3 \mid s_1, s_2, s_3 \in S\}$$. What is the smallest possible cardinality of $$T_A$$?

The best solution was submitted by 정희승 (서울대학교 물리천문학부, +4). Congratulations!

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (새내기과정학부 23학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 신민서(KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 최민규 (한양대학교 의학대학 졸업, +3), Eun Song (+3), James Hamilton Clerk (+3), Anar Rzayev (KAIST 전산학부 19학번, +2), 김준홍 (KAIST 수리과학과 20학번, +2), 최백규 (KAIST 수리과학과 석박통합과정 21학번, +3).

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# 2023-12 Pairs promoting diversity

Let $$p$$ be a prime number at least three and let $$k$$ be a positive integer smaller than $$p$$. Given $$a_1,\dots, a_k\in \mathbb{F}_p$$ and distinct elements $$b_1,\dots, b_k\in \mathbb{F}_p$$, prove that there exists a permutation $$\sigma$$ of $$[k]$$ such that the values of $$a_i + b_{\sigma(i)}$$ are distinct modulo $$p$$.

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Find all pairs of prime numbers $$(p, q)$$ such that $$pq$$ divides $$p^p + q^q + 1$$.