# 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!

Click here for his Solution of Problem 2009-2.

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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