# 2010-17 Two Hermitian Matrices

Let A, B be Hermitian matrices. Prove that tr(A2B2) ≥ tr((AB)2).

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# Break for the midterm exam

KAIST POW will take a break for the midterm exam. Good luck to all students!

Next problem will be posted on Oct. 29th.

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# Solution: 2010-16 Number of divisors in 1 (mod 3) or 2 (mod 3)

Let n be a positive integer. Let D(n,k) be the number of divisors x of n such that x≡k (mod 3). Prove that D(n,1)≥D(n,2).

The best solution was submitted by Jeong, Seong Gu (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2010-16.

Alternative solutions were submitted by 정진명 (수리과학과 2007학번, +3), 김치헌 (수리과학과 2006학번, +3), 박민재 (KSA-한국과학영재학교, +3).

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# 2010-16 Number of divisors in 1 (mod 3) or 2 (mod 3)

Let n be a positive integer. Let D(n,k) be the number of divisors x of n such that x≡k (mod 3). Prove that D(n,1)≥D(n,2).

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# Solution: 2010-15 Characteristic Polynomial

Let A, B be 2n×2n skew-symmetric matrices and let f be the characteristic polynomial of AB. Prove that the multiplicity of each root of f is at least 2.

The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-15.

An alternative solution was submitted by 정진명 (수리과학과 2007학번, +2).

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