# 2023-04 A perfect square

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

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# Notice on POW 2023-02

POW 2023-02 is still open. (Only a partial solution has been submitted.) Anyone who first submits a correct (full) solution will get the full credit.

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# 2023-03 Almost coverings of hypercubes

Determine the minimum number of hyperplanes in $$\mathbb{R}^n$$ that do not contain the origin but they together cover all points in $$\{0,1\}^n$$ except the origin.

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# Solution: 2023-01 An integral sequence (again)

Suppose $$a_1, a_2, \dots, a_{2023}$$ are real numbers such that
$a_1^3 + a_2^3 + \dots + a_n^3 = (a_1 + a_2 + \dots + a_n)^2$
for any $$n = 1, 2, \dots, 2023$$. Prove or disprove that $$a_n$$ is an integer for any $$n = 1, 2, \dots, 2023$$.

The best solution was submitted by 기영인 (KAIST 수리과학과 22학번, +4). Congratulations!

Other solutions were submitted by 고성훈 (KAIST 수리과학과 18학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 임도현 (KAIST 수리과학과 22학번, +3), 신정여 (KAIST 수리과학과 21학번, +3), 문강연 (KAIST 수리과학과 22학번, +3), 이명규 (KAIST 전산학과 20학번, +3), 박현영 (KAIST 전기및전자공학부 석박사통합과정 22학번, +3), Myint Mo Zwe (KAIST 새내기과정학부 22학번, +3), 이재경 (KAIST 뇌인지과학과 22학번, +3), Matthew Seok, 김기수 (KAIST 수리과학과 18학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), Yusuf Bahadir Kilicarslan (KAIST 전산학부 19학번, +3), 이동하 (KAIST 새내기과정학부 23학번, +2). Late solutions are not graded.

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# 2023-02 The fourth power of a real function

Let $$f(x)$$ be a degree 100 real polynomial. What is the largest possible number of negative coefficients of $$(f(x))^4$$?

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# 2023-01 An integral sequence (again)

Suppose $$a_1, a_2, \dots, a_{2023}$$ are real numbers such that
$a_1^3 + a_2^3 + \dots + a_n^3 = (a_1 + a_2 + \dots + a_n)^2$
for any $$n = 1, 2, \dots, 2023$$. Prove or disprove that $$a_n$$ is an integer for any $$n = 1, 2, \dots, 2023$$.

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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: 2022-23 The number of eigenvalues of 8 by 8 matrices

Let $$A$$ be an 8 by 8 integral unimodular matrix. Moreover, assume that for each $$x \in \mathbb{Z}^8$$, we have $$x^{\top} A x$$ is even. What is the possible number of positive eigenvalues for $$A$$?

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

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 여인영 (KAIST 물리학과 20학번, +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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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 $$S ~/\sim$$.