Prove that \(\displaystyle \sum_{m=0}^n \sum_{i=0}^m \binom{n}{m} \binom{m}{i}^3=\sum_{m=0}^n \binom{2m}{m} \binom{n}{m}^2\).

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Prove that \(\displaystyle \sum_{m=0}^n \sum_{i=0}^m \binom{n}{m} \binom{m}{i}^3=\sum_{m=0}^n \binom{2m}{m} \binom{n}{m}^2\).

KAIST POW will take a break for the midterm exam. Good luck to all students!

Next problem will be posted on March 26th.

Find the set of all rational numbers x such that 1, cos πx, and sin πx are linearly dependent over rational field Q.

The best solution was submitted by Chiheon Kim(김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-5.

An alternative solution was submitted by 정성구 (수리과학과 2007학번, +3), 임재원 (2009학번, +2). One incorrect solution was submitted.

Find the set of all rational numbers x such that 1, cos πx, and sin πx are linearly dependent over rational field Q.

Let n, k be positive integers. Prove that \(\sum_{i=1}^n k^{\gcd(i,n)}\) is divisible by n.

The best solution was submitted by Chiheon Kim(김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-4.

Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), Prach Siriviriyakul (2009학번, +3), 라준현 (수리과학과 2008학번, +3), 서기원 (2009학번, +3), 강동엽 (2009학번, +3), 임재원 (2009학번, +2).