Define a sequence \( a_n \) by \( a_1 = 1 \) and
\[
a_{n+1} = \frac{1}{n} \left( 1 + \sum_{k=1}^n a_k^2 \right)
\]
for any \( n \geq 1 \). Prove or disprove that \( a_n \) is an integer for all \( n \geq 1 \).

The best solution was submitted by 채지석 (KAIST 수리과학과 석박통합과정, +4). Congratulations!

Here is the best solution of problem 2022-22.

Other solutions were submitted by 기영인 (KAIST 22학번, +3), 김기수 (KAIST 수리과학과 18학번, +3), 박준성 (KAIST 수리과학과 석박통합과정, +3). An incomplete solution was submitted.

Let \(S\) be the set of all 4 by 4 integral positive-definite symmetric unimodular matrices. Define an equivalence relation \( \sim \) on \(S\) such that for any \( A,B \in S\), we have \(A \sim B\) if and only if \(PAP^\top = B\) for some integral unimodular matrix \(P\). Determine \(S ~/\sim \).

The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!

Here is the best solution of problem 2022-20.

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!

Here is the best solution of problem 2022-19.

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).

Let \(a(n)\) be the number of unordered factorizations of \(n\) into divisors larger than \(1\). Prove that \(\sum_{n=2}^{\infty} \frac{a(n)}{n^2} = 1\).

The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!

Here is the best solution of problem 2022-18.

Other solutions were submitted by 기영인 (KAIST 22학번, +3), Kawano Ren (Kaisei Senior High School, +3), Sakae Fujimoto (Osaka Prefectural Kitano High School, Freshmen, +3), 최백규 (KAIST 생명과학과 20학번, +3).

Let \(n, i\) be integers such that \(1 \leq i \leq n\). Each subset of \( \{ 1, 2, \ldots, n \} \) with \( i\) elements has the smallest number. We define \( \phi(n,i) \) to be the sum of these smallest numbers. Compute \[ \sum_{i=1}^n \phi(n,i).\]

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

Here is the best solution of problem 2022-17.

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 기영인 (KAIST 22학번, +3), 이준환 (한국외국어대학교 통계학과 19학번, +3), 오준혁 (KAIST 수리과학과 20학번, +3), 신준범 (컬럼비아 대학교 20학번, +3), 이한스 (KAIST 수리과학과 20학번, +3), Kawano Ren (Kaisei Senior High School, +3), Sakae Fujimoto (Osaka Prefectural Kitano High School, Freshmen, +3).

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 \[ \det\left( \begin{matrix} S(m+1,1) & S(m+1,2) & \cdots & S(m+1,n) \\
S(m+2,1) & S(m+2,2) & \cdots & S(m+2,n) \\
\cdots & \cdots & \cdots & \cdots \\
S(m+n,1) & S(m+n,2) & \cdots & S(m+n,n) \end{matrix} \right) = (n!)^m \]

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

Here is the best solution of problem 2022-15.

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 김찬우 (연세대학교 수학과, +3). Late solutions were not graded.

For a positive integer \(n\), let \(B\) and \(C\) be real-valued \(n\) by \(n\) matrices and \(O\) be the \(n\) by \(n\) zero matrix. Assume further that \(B\) is invertible and \(C\) is symmetric. Define \[A := \begin{pmatrix} O & B \\ B^T & C \end{pmatrix}.\] What is the possible number of positive eigenvalues for \(A\)?

The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!

Here is the best solution of problem 2022-14.

Prove for any \( x \geq 1 \) that

\[
\left( \sum_{n=0}^{\infty} (n+x)^{-2} \right)^2 \geq 2 \sum_{n=0}^{\infty} (n+x)^{-3}.
\]

The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!

Here is the best solution of problem 2022-13.

Another solution was submitted by 김찬우 (연세대학교 수학과, +3).