Daily Archives: March 16, 2012

2012-6 Matrix modulo p

Let p be a prime number and let n be a positive integer. Let \(A=\left( \binom{i+j-2}{i-1}\right)_{1\le i\le p^n, 1\le j\le p^n} \) be a \(p^n \times p^n\) matrix. Prove that \( A^3 \equiv I \pmod p\), where I is the \(p^n \times p^n\) identity matrix.

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Solution: 2012-5 Iterative geometric mean

For given positive real numbers \(a_1,\ldots,a_k\) and for each integer n≥k, let \(a_{n+1}\) be the geometric mean of \( a_n, a_{n-1}, a_{n-2}, \ldots, a_{n-k+1}\). Prove that \( \lim_{n\to\infty} a_n\) exists and compute this limit.

The best solution was submitted by Gee Won Suh (서기원), 수리과학과 2009학번. Congratulations!

Here is his Solution of Problem 2012-5.

Alternative solutions were submitted by 박민재 (2011학번, +3, Solution), 김태호 (2011학번, +3, Solution), 이명재 (2012학번, +3), 박훈민 (대전과학고등학교 2학년, +3), 윤영수 (2011학번, +2), 조준영 (2012학번, +2), 변성철 (2011학번, +2), 정우석 (서강대학교 자연과학부 2011학번, +2). One incorrect solution was received.

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