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.
Let S be the set of non-zero real numbers x such that there is exactly one 0-1 sequence {an} satisfying \(\displaystyle \sum_{n=1}^\infty a_n x^{-n}=1\). Prove that there is a one-to-one function from the set of all real numbers to S.
The best solution was submitted by Jeong, Seong Gu (정성구), 수리과학과 2007학번. Congratulations!
Here is his Solution of Problem 2010-7.