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

**GD Star Rating**

*loading...*

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

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

Next problem will be posted on Oct. 29th.

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).

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).

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).

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.