Category Archives: solution

Solution: 2022-21 A determinant of greatest common divisors

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 \(\det(A) = \prod_{i=1}^{n} \varphi(a_i)\).

The best solution was submitted by Noitnetta Yobepyh (Snaejwen High School, +4). Congratulations!

Here is the best solution of problem 2022-21.

Other solutions were submitted by 기영인 (KAIST 22학번, +3), 여인영 (KAIST 물리학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정, +3), 전해구 (KAIST 기계공학과 졸업생, +2), 최예준 (서울과기대 행정학과 21학번, +2).

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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!

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).

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Solution: 2022-18 A sum of the number of factorizations

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).

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Solution: 2022-17 The smallest number of subsets

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).

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Solution: 2022-16 Identity for continuous functions

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.

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Solution: 2022-15 A determinant of Stirling numbers of second kind

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.

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Solution: 2022-14 The number of eigenvalues of a symmetric matrix

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.

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Solution: 2022-13 Inequality involving sums with different powers

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).

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Solution: 2022-12 A partition of the power set of a set

Consider the power set \(P([n])\) consisting of \(2^n\) subsets of \([n]=\{1,\dots,n\}\).
Find the smallest \(k\) such that the following holds: there exists a partition \(Q_1,\dots, Q_k\) of \(P([n])\) so that there do not exist two distinct sets \(A,B\in P([n])\) and \(i\in [k]\) with \(A,B,A\cup B, A\cap B \in Q_i\).

The best solution was submitted by 조유리 (문현여고 3학년, +4). Congratulations!

Here is the best solution of problem 2022-12.

Other solutions were submitted by 박기찬 (KAIST 새내기과정학부 22학번, +3), 김기수 (KAIST 수리과학과 18학번, +3), 신준범 (컬럼비아 대학교 20학번, +3), 이종서 (KAIST 전산학부 19학번, +3).

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