# Solution: 2023-04 A perfect square

Find all integers $$n$$ such that $$n^4 + n^3 + n^2 + n + 1$$ is a perfect square.

The best solution was submitted by 채지석 (KAIST 수리과학과 석박사통학과정 21학번, +4). Congratulations!

Other solutions were submitted by 기영인 (KAIST 수리과학과 22학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 노희윤 (KAIST 수리과학과 19학번, +3), 문강연 (KAIST 수리과학과 22학번, +3), 이명규 (KAIST 전산학과 20학번, +3), 박지환 (연세대학교 수학과 22학번, +3), 백민수 (원주중학교 교사, +3), 이종서 (KAIST 전산학부 19학번, +3), Matthew Seok, 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), 이동하 (KAIST 새내기과정학부 23학번, +3).

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# Solution: 2022-24 Hey, who turned out the lights?

There are light bulbs $$\ell_1,\dots, \ell_n$$ controlled by the switches $$s_1, \dots, s_n$$. The $$i$$th switch flips the status of the $$i$$th light and possibly others as well. If $$s_i$$ flips the status of $$\ell_j$$, then $$s_j$$ flips the status of $$\ell_i$$. All lights are initially off. Prove that it is possible to turn all the lights on.

The best solution was submitted by 채지석 (KAIST 수리과학과 석박통합과정, +4). Congratulations!

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 박준성 (KAIST 수리과학과 석박통합과정, +3).

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# 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.

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# Solution: 2020-16 A convex function of matrices

Let $$A$$ be an $$n \times n$$ Hermitian matrix and $$\lambda_1 (A) \geq \lambda_2 (A) \geq \dots \geq \lambda_n (A)$$ the eigenvalues of $$A$$. Prove that for any $$1 \leq k \leq n$$
$A \mapsto \lambda_1 (A) + \lambda_2 (A) + \dots + \lambda_k (A)$
is a convex function.

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-16.

Other solutions were submitted by 길현준 (수리과학과 2018학번, +3), 이준호 (수리과학과 2016학번, +3).

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# Solution: 2020-14 Connecting dots probabilistically

Say there are n points. For each pair of points, we add an edge with probability 1/3. Let $$P_n$$ be the probability of the resulting graph to be connected (meaning any two vertices can be joined by an edge path). What can you say about the limit of $$P_n$$ as n tends to infinity?

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-14.

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 김건우 (수리과학과 2017학번, +3), 이준호 (수리과학과 2016학번, +3), 김유일 (2020학번, +3).

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# Solution: 2020-12 Draws on a chess tournament

There are $$n$$ people participating to a chess tournament and every two players play exactly one game against each other. The winner receives $$1$$ point and the loser gets $$0$$ point and if the game is a draw, each player receives $$0.5$$ points. Prove that if at least $$3/4$$ of the games are draws, then there are two players with the same total scores.

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-12.

Another solution was submitted by 고성훈 (수리과학과 2018학번, +3).

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# Solution: 2020-11 Free group of rank 2

Show that there is a subgroup of a free group of ran 2 that is not finitely generated.

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-11.

Other solutions were submitted by 조한슬 (수리과학과 2017학번, +3), 최백규(생명과학과 대학원, +2).

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# Solution: 2019-19 Balancing consecutive squares

Find all integers $$n$$ such that the following holds:

There exists a set of $$2n$$ consecutive squares $$S = \{ (m+1)^2, (m+2)^2, \dots, (m+2n)^2 \}$$ ($$m$$ is a nonnegative integer) such that $$S = A \cup B$$ for some $$A$$ and $$B$$ with $$|A| = |B| = n$$ and the sum of elements in $$A$$ is equal to the sum of elements in $$B$$.

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2019-19.

An incorrect solution was submitted.

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# Solution: 2019-15 Singular matrix

Let $$A, B$$ be $$n \times n$$ Hermitian matrices. Find all positive integer $$n$$ such that the following statement holds:

“If $$AB – BA$$ is singular, then $$A$$ and $$B$$ have a common eigenvector.”

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2019-14.

A similar solution was submitted by 하석민 (수리과학과 2017학번, +3). Late solutions are not graded.

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Suppose that $$f: \mathbb{R} \to \mathbb{R}$$ is differentiable and $$\max_{ x \in \mathbb{R}} |f(x)| = M < \infty$$. Prove that $\int_{-\infty}^{\infty} (|f'|^2 + |f|^2) \geq 2M^2.$