# Solution: 2009-6 Sum of integers of the fourth power

Prove that each positive integer x can be written as a sum of at most 53 integers to the fourth power. In other words, for every positive integer x, there exist $$y_1,y_2,\ldots,y_k$$ with $$k\le 53$$ such that $$x=\sum_{i=1}^k y_i^4$$.

The best solution was submitted by SangHoon Kwon (권상훈), 수리과학과 2006학번.

Here is his Solution of Problem 2009-6.

There were 4 other submitted solutions: 백형렬, 이재송, 조강진, 김치헌 (+3).

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# Solution: 2009-2 Sequence of Log

Let $$a_1<\cdots$$ be a sequence of positive integers such that $$\log a_1, \log a_2,\log a_3,\cdots$$ are linearly independent over the rational field $$\mathbb Q$$. Prove that $$\lim_{k\to \infty} a_k/k=\infty$$.

The best solution was submitted by SangHoon Kwon (권상훈), 수리과학과 2006학번. Congratulations!

There were 3 other submitted solutions which will earn points: 김치헌+3, 김린기+3,  조강진+2.

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# Solution: 2008-10 Inequality with n variables

Let $$x_1,x_2,\ldots,x_n$$ be nonnegative real numbers. Show that
$$\displaystyle \left(\sum_{i=1}^n x_i\right) \left(\sum_{i=1}^n x_i^{n-1}\right) \le (n-1) \sum_{i=1}^n x_i^n + n \prod_{i=1}^n x_i$$.

The best solution was submitted by Sang Hoon Kwon (권상훈), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2008-10.

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# Solution: 2008-8 Positive eigenvalues

Let A be a 0-1 square matrix. If all eigenvalues of A are real positive, then those eigenvalues are all equal to 1.

The best solution was submitted by Sang Hoon Kwon (권상훈), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2008-8.

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