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.
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.
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 \(A\)?
Let \(\varphi(x)\) be the Euler’s totient function. Let \(S = \{a_1,\dots, a_n\}\) be a set of positive integers such that for any \(a_i\), all of its positive divisors are also in \(S\). Let \(A\) be the matrix with entries \(A_{i,j} = gcd(a_i,a_j)\) being the greatest common divisors of \(a_i\) and \(a_j\). Prove that \(\det(A) = \prod_{i=1}^{n} \varphi(a_i)\).
The best solution was submitted by Noitnetta Yobepyh (Snaejwen High School, +4). Congratulations!
Here is the best solution of problem 2022-21.
Other solutions were submitted by 기영인 (KAIST 22학번, +3), 여인영 (KAIST 물리학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정, +3), 전해구 (KAIST 기계공학과 졸업생, +2), 최예준 (서울과기대 행정학과 21학번, +2).
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 \( n \geq 1 \).
POW 2022-20 is still open. Anyone who first submits a correct solution will get the full credit.