# Concluding Spring 2013

The top 5 participants of the semester are:

• 1st: 라준현 (08학번): 38 points
• 2nd: 서기원 (09학번): 29 points
• T-3rd: 김호진 (09학번): 25 points
• T-3rd: 황성호 (13학번): 25 points
• 5th: 김범수 (10학번): 19 points

Hearty congratulations to the prize winners! The prize ceremony will be held on Jun. 19 (Wed.) at 2PM.

We thank all of the participants for the nice solutions and your intereset you showed for POW. We hope to see you next semester with even better problems.

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# Solution: 2013-07 Maximum number of points

Consider the unit sphere in $$\mathbb{R}^n$$. Find the maximum number of points on the sphere such that the (Euclidean) distance between any two of these points is larger than $$\sqrt 2$$.

The best solution was submitted by 라준현, 08학번. Congratulations!

Other solutions were submitted by 서기원(09학번, +3), 황성호(13학번, +3), 김범수(10학번, +3), 전한솔(고려대, +3), 홍혁표(13학번, +2), 어수강(서울대, +2). Thank you for your participation.

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Let $$A, B$$ be $$N \times N$$ symmetric matrices with eigenvalues $$\lambda_1^A \leq \lambda_2^A \leq \cdots \leq \lambda_N^A$$ and $$\lambda_1^B \leq \lambda_2^B \leq \cdots \leq \lambda_N^B$$. Prove that
$\sum_{i=1}^N |\lambda_i^A – \lambda_i^B|^2 \leq Tr (A-B)^2$