# 2009-7 A rational problem

Let n>1 be an integer and let x>1 be a real number. Prove that if
$$\sqrt[n]{x+\sqrt{x^2-1}}+\sqrt[n]{x-\sqrt{x^2-1}}$$
is a rational number, then x is rational.

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# No problem on March 20

Due the midterm exam, no new problems will be posted on March 20. We will continue on March 27.

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# 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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# 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 $$x=\sum_{i=1}^k y_i^4$$.

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# Solution: 2009-5 Random points and the origin

If we choose n points on a circle randomly and independently with uniform distribution, what is the probability that the center of the circle is contained in the interior of the convex hull of these n points?

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

There were 2 incorrect solutions submitted.

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# 2009-5 Random points and the origin

If we choose n points on a circle randomly and independently with uniform distribution, what is the probability that the center of the circle is contained in the interior of the convex hull of these n points?

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Let $$a_0=a$$ and $$a_{n+1}=a_n (a_n^2-3)$$. Find all real values $$a$$ such that the sequence $$\{a_n\}$$ converges.