Problem 2009-23 was the last problem of this semester. Best wishes and good luck to your final exam. We wish you to come back in the spring semester next year; we will start in the first week of February.
Acknowledgment:
Two problems 2009-14 and 2009-15, were contributed by Sungyoon Kim (김성윤), a graduate student in MIT. Thanks.
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.
The best solution was submitted by Sungyoon Kim (김성윤) (Mathematics, MIT, Class of ’09). Congratulations! (Though, he is not eligible for earning points and taking prizes.)
Here is his Solution of Problem 2009-7.
There were 5 other solutions submitted: 김호진 (2009학번), 백형렬 (수리과학과 2003학번), 이재송 (전산학과 2005학번), 조강진 (2009학번), 박승균 (수리과학과 2008학번). All will receive 3 points each.