# 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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# Solution: 2022-20 4 by 4 symmetric integral matrices

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!

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# 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.

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# 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 $$A$$?

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# Solution: 2022-21 A determinant of greatest common divisors

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!

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).

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# 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 $$n \geq 1$$.

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# Notice on POW 2022-20

POW 2022-20 is still open. Anyone who first submits a correct solution will get the full credit.

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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 $$\det(A) = \prod_{i=1}^{n} \varphi(a_i)$$.