Daily Archives: March 9, 2012

Solution: 2012-4 Sum of squares

Find the smallest and the second smallest odd integers n satisfying the following property: \[ n=x_1^2+y_1^2 \text{ and } n^2=x_2^2+y_2^2 \] for some positive integers \(x_1,y_1,x_2,y_2\) such that \(x_1-y_1=x_2-y_2\).

The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!

Here is his Solution of Problem 2012-4.

Alternative solutions were submitted by 조준영 (2012학번, +3), 서기원 (수리과학과 2009학번, +3), 임창준 (2012학번, +3), 홍승한 (2012학번, +2), 이명재 (2012학번, +2), 김현수 (?, +3), 천용 (전남대, +2). One incorrect solution was received.

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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.

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