# 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 $$T_A$$?

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# Solution: 2023-09 Permuted sums of reciprocals

Let $$\mathbb{S}_n$$ be the set of all permutations of $$[n]=\{1,\dots, n\}$$. For positive real numbers $$d_1,\dots, d_n$$, prove $\sum_{\sigma\in \mathbb{S}_n} \frac{1}{ d_{\sigma(1)}(d_{\sigma(1)}+d_{\sigma(2)}) \dots (d_{\sigma(1)}+\dots + d_{\sigma(n)}) } = \frac{1}{d_1\dots d_n}.$

The best solution was submitted by 신민서 (KAIST 수리과학과 20학번, +4). Congratulations!

Other solutions were submitted by 권도현 (KAIST 수리과학과 22학번, +3), 김명규 (KAIST 전산학부 19학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3),이명규 (KAIST 전산학부 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3). James Hamilton Clerk (+3), Matthew Seok (+3).

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# 2023-10 A pair of primes

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

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# Solution: 2023-08 Groups with a perfect commutator subgroup

Find a pair of nonisomorphic nonabelian groups so that their abelianizations are isomorphic and their commutator subgroups are perfect.

The best solution was submitted by 김찬우 (연세대학교 수학과 22학번, +4). Congratulations!

Other solutions were submitted by 박기윤 (KAIST 새내기과정학부 23학번, +3), 이명규 (KAIST 전산학과 20학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +2).

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# 2023-09 Permuted sums of reciprocals

Let $$\mathbb{S}_n$$ be the set of all permutations of $$[n]=\{1,\dots, n\}$$. For positive real numbers $$d_1,\dots, d_n$$, prove $\sum_{\sigma\in \mathbb{S}_n} \frac{1}{ d_{\sigma(1)}(d_{\sigma(1)}+d_{\sigma(2)}) \dots (d_{\sigma(1)}+\dots + d_{\sigma(n)}) } = \frac{1}{d_1\dots d_n}.$

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# Solution: 2023-07 An oscillatory integral

Suppose that $$f: [a, b] \to \mathbb{R}$$ is a smooth, convex function, and there exists a constant $$t>0$$ such that $$f'(x) \geq t$$ for all $$x \in (a, b)$$. Prove that
$\left| \int_a^b e^{i f(x)} dx \right| \leq \frac{2}{t}.$

The best solution was submitted by Anar Rzayev (KAIST 전산학부 19학번, +4). Congratulations!

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 오현섭 (KAIST 수리과학과 박사과정 21학번, +3), 이명규 (KAIST 전산학과 20학번, +3), 최예준 (서울과학기술대학교 행정학과 21학번, +3), Matthew Seok (+3), James Hamilton Clerk (+3).

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