# 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).

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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!

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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!

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

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# Solution: 2022-11 groups with torsions

Does there exists a finitely generated group which contains torsion elements of order p for all prime numbers p?

The best solution was submitted by 김기수 (KAIST 수리과학과 2018학번, +4). Congratulations!

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# Solution: 2022-08 two sequences

For positive integers $$n \geq 2$$, let $$a_n = \lceil n/\pi \rceil$$ and let $$b_n = \lceil \csc (\pi/n) \rceil$$. Is $$a_n = b_n$$ for all $$n \neq 3$$?

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

Other solutions were submitted by 조유리 (문현여고 3학년, +3), 이명규 (KAIST 전산학부 20학번, +3), 박기찬 (KAIST 새내기과정학부 22학번, +3), 여인영 (KAIST 물리학과 20학번, +3).

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# Solution: 2021-05 Finite generation of a group

Prove or disprove that if all elements of an infinite group G has order less than n for some positive integer n, then G is finitely generated.

The best solution was submitted by 김기수 (수리과학과 2018학번, +4). Congratulations!

Here is his solution of problem 2021-05.

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 고성훈 (수리과학과 2018학번, +3), Late solutions are not graded.

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# Solution: 2020-05 Completion of a metric space

We say a metric space complete if every Cauchy sequence converges.

Let (X, d) be a metric space. Show that there exists an isometric imbedding from X to a complete metric space Y so that the image of X in Y is dense.

The best solution was submitted by 김기수 (수리과학과 2018학번). Congratulations!

Here is his solution of problem 2020-05.

Other solutions were submitted by 고성훈 (수리과학과 2018학번, +3), 구은한 (수리과학과 2019학번, +3), 길현준 (수리과학과 2018학번, +3), 김기택 (수리과학과 2015학번, +3), 이준호 (2016학번, +3).

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