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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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.

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

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).

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 \(x=\sum_{i=1}^k y_i^4\).

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.

Click here for his Solution of Problem 2009-5.

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?

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.

The best solution was submitted by Jaesong Lee (이재송), 전산학과 2005학번. Congratulations!

Check his Solution of Problem 2009-4. (This proof can be slightly improved in the second half.)

Alternative solutions were submitted by 김치헌 (+2), 백형렬 (+3).