# Solution: 2023-01 An integral sequence (again)

Suppose $$a_1, a_2, \dots, a_{2023}$$ are real numbers such that
$a_1^3 + a_2^3 + \dots + a_n^3 = (a_1 + a_2 + \dots + a_n)^2$
for any $$n = 1, 2, \dots, 2023$$. Prove or disprove that $$a_n$$ is an integer for any $$n = 1, 2, \dots, 2023$$.

The best solution was submitted by 기영인 (KAIST 수리과학과 22학번, +4). Congratulations!

Other solutions were submitted by 고성훈 (KAIST 수리과학과 18학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 임도현 (KAIST 수리과학과 22학번, +3), 신정여 (KAIST 수리과학과 21학번, +3), 문강연 (KAIST 수리과학과 22학번, +3), 이명규 (KAIST 전산학과 20학번, +3), 박현영 (KAIST 전기및전자공학부 석박사통합과정 22학번, +3), Myint Mo Zwe (KAIST 새내기과정학부 22학번, +3), 이재경 (KAIST 뇌인지과학과 22학번, +3), Matthew Seok, 김기수 (KAIST 수리과학과 18학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), Yusuf Bahadir Kilicarslan (KAIST 전산학부 19학번, +3), 이동하 (KAIST 새내기과정학부 23학번, +2). Late solutions are not graded.

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# Solution: 2022-19 Inequality for twice differentiable functions

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a twice differentiable function satisfying $$f(0) = 0$$ and $$0 \leq f'(x) \leq 1$$. Prove that
$\left( \int_0^1 f(x) dx \right)^2 \geq \int_0^1 [f(x)]^3 dx.$

The best solution was submitted by 기영인 (KAIST 22학번, +4). Congratulations!

Other solutions were submitted by 여인영 (KAIST 물리학과 20학번, +3), Kawano Ren (Kaisei Senior High School, +3), 최예준 (서울과기대 행정학과 21학번, +3), 김준성 (KAIST 물리학과 박사과정, +3).

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Let $$S(n,k)$$ be the Stirling number of the second kind that is the number of ways to partition a set of $$n$$ objects into $$k$$ non-empty subsets. Prove the following equality $\det\left( \begin{matrix} S(m+1,1) & S(m+1,2) & \cdots & S(m+1,n) \\ S(m+2,1) & S(m+2,2) & \cdots & S(m+2,n) \\ \cdots & \cdots & \cdots & \cdots \\ S(m+n,1) & S(m+n,2) & \cdots & S(m+n,n) \end{matrix} \right) = (n!)^m$