Let \(\varphi(x)\) be the Euler’s totient function. Let \(S = \{a_1,\dots, a_n\}\) be a set of positive integers such that for any \(a_i\), all of its positive divisors are also in \(S\). Let \(A\) be the matrix with entries \(A_{i,j} = gcd(a_i,a_j)\) being the greatest common divisors of \(a_i\) and \(a_j\). Prove that \(\det(A) = \prod_{i=1}^{n} \varphi(a_i)\).
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 \(S\) 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 \(S ~/\sim \).
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
\[ \left( \int_0^1 f(x) dx \right)^2 \geq \int_0^1 [f(x)]^3 dx. \]
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).
POW will resume on Oct. 28.
Let \(a(n)\) be the number of unordered factorizations of \(n\) into divisors larger than \(1\). Prove that \(\sum_{n=2}^{\infty} \frac{a(n)}{n^2} = 1\).
For a positive integer \( n \), find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) such that
\[
\sum_{k=0}^n \binom{n}{k} f(x^{2^k}) = 0
\]
for all \( x \in \mathbb{R} \).
The best solution was submitted by Kawano Ren (Kaisei Senior High School, +4). Congratulations!
Here is the best solution of problem 2022-16.
Other solutions were submitted by 김찬우 (연세대학교 수학과, +3), 기영인 (KAIST 22학번, +3), 여인영 (KAIST 물리학과 20학번, +3). Late solutions were not graded.
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 \[ \sum_{i=1}^n \phi(n,i).\]