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

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

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.

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.

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!

Here is the best solution of problem 2023-01.

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.

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

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 \( n = 1, 2, \dots, 2023 \).

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!

Here is the best solution of problem 2022-24.

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

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!

Here is the best solution of problem 2022-23.

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 여인영 (KAIST 물리학과 20학번, +3).

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!

Here is the best solution of problem 2022-22.

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

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!

Here is the best solution of problem 2022-20.