# Solution: 2022-22 An integral sequence

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!

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

GD Star Rating

# Solution: 2022-20 4 by 4 symmetric integral matrices

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!

GD Star Rating

# 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!

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).

GD Star Rating

# 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).

GD Star Rating

# 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!

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).

GD Star Rating

# 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!

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).

GD Star Rating

# 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!

Other solutions were submitted by 김찬우 (연세대학교 수학과, +3), 기영인 (KAIST 22학번, +3), 여인영 (KAIST 물리학과 20학번, +3). Late solutions were not graded.

GD Star Rating

# 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!

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

GD Star Rating

# 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!

GD Star Rating
Prove for any $$x \geq 1$$ that
$\left( \sum_{n=0}^{\infty} (n+x)^{-2} \right)^2 \geq 2 \sum_{n=0}^{\infty} (n+x)^{-3}.$