# Solution: 2023-20 A sequence with small tail

Can we find a sequence $$a_i, i=0,1,2,…$$ with the following property: for each given integer $$n\geq 0$$, we have $\lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} |a_i|\leq 23^{(n+11)^{10}} \quad \text{ and }\quad \lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} a_i = (-1)^n ?$

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

Another solution was submitted by 조현준 (KAIST 수리과학과 22학번, +2).

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# Solution: 2023-15 An inequality for complex polynomials

Let $$p(z), q(z)$$ and $$r(z)$$ be polynomials with complex coefficients in the complex plane. Suppose that $$|p(z)| + |q(z)| \leq |r(z)|$$ for every $$z$$. Show that there exist two complex numbers $$a,b$$ such that $$|a|^2 +|b|^2 =1$$ and $$a p(z) + bq(z) =0$$ for every $$z$$.

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

Other solutions were submitted by 강지민 (세마고 3학년, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 신민서 (KAIST 수리과학과 20학번, +3), 여인영 (KAIST 물리학과 20학번, +3),이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 조현준 (KAIST 수리과학과 22학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 최민규 (한양대학교 의과대학 졸업생, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), Muhammadfiruz Hasanov (+3).

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# 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!

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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!

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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