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

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# Solution: 2022-19 Inequality for twice differentiable functions

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a twice differentiable function satisfying $$f(0) = 0$$ and $$0 \leq f'(x) \leq 1$$. Prove that
$\left( \int_0^1 f(x) dx \right)^2 \geq \int_0^1 [f(x)]^3 dx.$

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

Other solutions were submitted by 여인영 (KAIST 물리학과 20학번, +3), Kawano Ren (Kaisei Senior High School, +3), 최예준 (서울과기대 행정학과 21학번, +3), 김준성 (KAIST 물리학과 박사과정, +3).

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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 $$S ~/\sim$$.