Evaluate the sum \[ \sum_{n=1}^{\infty} \frac{n \sin n}{1+n^2}. \]
The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!
Here is his Solution of Problem 2011-1.
A solution for an incorrect version of the problem was submitted by 구도완 (해운대고등학교 3학년).
Let \(a_1=0\), \(a_{2n+1}=a_{2n}=n-a_n\). Prove that there exists k such that \(\lvert a_k- \frac{k}{3}\rvert > 2010\) and yet \(\lim_{n\to \infty} \frac{a_n}{n}=\frac13\).
The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!
Here is his Solution of Problem 2010-21.
Alternative solutions were submitted by 한대진 (신현여중 교사, +2), 이승훈 (연세대학교 경제학과 06학번, +2).
Let X be a finite set of points on the plane such that each point in X is colored with red or blue and there is no line having all points in X. Prove that there is a line L having at least two points of X such that all points in L∩X have the same color.
The best solution was submitted by Minjae Park (박민재), 한국과학영재학교 (KSA). Congratulations!
Here is his Solution of Problem 2010-20.
Suppose that \(V\) is a vector space of dimension \(n>0\) over a field of characterstic \(p\neq 0\). Let \(A: V\to V\) be an affine transformation. Prove that there exist \(u\in V\) and \(1\le k\le np\) such that \[A^k u = u.\]
The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!
Here is his Solution of Problem 2010-19.
An alternative solution was submitted by 박민재 (KSA-한국과학영재학교, +3).
Let f be a differentiable function. Prove that if \(\lim_{x\to\infty} (f(x)+f'(x))=1\), then \(\lim_{x\to\infty} f(x)=1\).
The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!
Here is his Solution of Problem 2010-18.
Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), 서기원 (수리과학과 2009학번, +3), 심규석 (수리과학과 2007학번, +3), 진우영 (KSA-한국과학영재학교, +3), 박민재 (KSA-한국과학영재학교, +2), 한대진 (?, +2), 문정원 (성균관대학교 수학교육과, +2).
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**).