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.

• Solution by 오성진 using the Cauchy integral formula.
• Solution by 정진명 NOT using the binomial theorem.
• Two solutions by 박민재 : (1) A combinatorial solution, (2) Using generating functions and Lagrange’s Inversion Theorem.

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.