Let A, B be Hermitian matrices. Prove that tr(A2B2) ≥ tr((AB)2).
The best solution was submitted by Jeong, Jinmyeong (정진명), 수리과학과 2007학번. Congratulations!
Here is his Solution of Problem 2010-17.
Alternative solutions were submitted by 정성구 (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).
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 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 n be a positive integer. Prove that
\(\displaystyle \sum_{k=0}^n (-1)^k \binom{2n+2k}{n+k} \binom{n+k}{2k}=(-4)^n\).
The best solution was submitted by Gee Won Suh (서기원), 2009학번. Congratulations!
Here is his Solution of Problem 2010-14.
Alternative solutions were submitted by 김치헌 (수리과학과 2006학번, +3), 정진명 (수리과학과 2007학번, +3), 박민재 (KSA-한국과학영재학교, +3), 오성진 (Princeton Univ.), Abhishek Verma (GET-SKEC NDEC, New Delhi).
Here are some interesting solutions.
Prove that there is a constant C such that
\(\displaystyle \sup_{A<B} \int_A^B \sin(x^2+ yx) \, dx \le C\)
for all y.
The best solution was submitted by Minjae Park (박민재), KSA (한국과학영재학교) 3학년. Congratulations!
Here is his Solution of Problem 2010-13.
Alternative solutions were submitted by 정진명 (수리과학과 2007학번, +3), 정성구 (수리과학과 2007학번, +3), 심규석 (수리과학과 2007학번, +3). Three incorrect solutions were submitted (서**, 정**, Ver**).
Let A be a square matrix. Prove that there exists a diagonal matrix J such that A+J is invertible and each diagonal entry of J is ±1.
The best solution was submitted by Jeong, Jinmyeong (정진명), 수리과학과 2007학번. Congratulations!
Here is his Solution of Problem 2010-12.
Alternative solutions were submitted by 권용찬 (수리과학과 2009학번, +3), 심규석 (수리과학과 2007학번, +3), 정성구 (수리과학과 2007학번, +3), 정유중 (2006학번, +3), 김치헌 (수리과학과 2006학번, +3), 박민재 (KSA-한국과학영재학교, +3), 서영우 (2010학번, +2), 서기원 (2009학번, +2), 오상국 (2007학번, +2). One of them has a non-constructive solution of Problem 2010-12.
Let z be a real number. Find all solutions of the following integral equation: \(f(x)=e^x+z \int_0^1 e^{x-y} f(y)\,dy\) for 0≤x≤1.
The best solution was submitted by Gee Won Suh (서기원), 2009학번. Congratulations!
Here is his Solution of Problem 2010-11.
Alternative solutions were submitted by 최홍석 (화학과 2006학번, +3), 정성구 (수리과학과 2007학번, +3).
Let Mn×n be the space of real n×n matrices, regarded as a metric space with the distance function
\(\displaystyle d(A,B)=\sum_{i,j} |a_{ij}-b_{ij}|\)for A=(aij) and B=(bij).
Prove that \(\{A\in M_{n\times n}: A^m=0 \text{ for some positive integer }m\}\) is a closed set.
The best solution was submitted by Gee Won Suh (서기원), 2009학번. Congratulations!
Here is his Solution of Problem 2010-10.
Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), 김치헌 (수리과학과 2006학번, +3), 강동엽 (2009학번, +2).
Let M>0 be a real number. Prove that there exists N so that if n>N, then all the roots of \(f_n(z)=1+\frac{1}{z}+\frac1{{2!}z^2}+\cdots+\frac{1}{n!z^n}\) are in the disk |z|<M on the complex plane.
The best solution was submitted by Jeong, Seong Gu (정성구), 수리과학과 2007학번. Congratulations!
Here is his Solution of Problem 2010-9.
Alternative solutions were submitted by 최홍석(화학과 2006학번, +3), 김호진(2009학번, +3), 김치헌 (수리과학과 2006학번, +3).
Prove that \(\displaystyle \sum_{m=0}^n \sum_{i=0}^m \binom{n}{m} \binom{m}{i}^3=\sum_{m=0}^n \binom{2m}{m} \binom{n}{m}^2\).
The best solution was submitted by Chiheon Kim(김치헌), 수리과학과 2006학번. Congratulations!
Here is his Solution of Problem 2010-6.