Let \(\{x_1, x_2, \ldots, x_{21}\} = \{-10, -9, \ldots, -1, 0, 1, \ldots, 9, 10\}\). What is the largest possible value of \(x_1x_2x_3+x_4x_5x_6+\cdots + x_{19}x_{20}x_{21}\)?

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Let \(\{x_1, x_2, \ldots, x_{21}\} = \{-10, -9, \ldots, -1, 0, 1, \ldots, 9, 10\}\). What is the largest possible value of \(x_1x_2x_3+x_4x_5x_6+\cdots + x_{19}x_{20}x_{21}\)?

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.

The best solution was submitted by 이종서 (KAIST 전산학부 19학번, +4). Congratulations!

Here is the best solution of problem 2023-03.

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3). There were two incorrect solutions submitted.

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