Prove that \(\sqrt{2}+\sqrt[3]{5}\) is irrational.

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Prove that \(\sqrt{2}+\sqrt[3]{5}\) is irrational.

Evaluate the following limit:

\(\displaystyle \lim_{\varepsilon\to 0}\int_0^{2\varepsilon} \log\left(\frac{|\sin t-\varepsilon|}{\sin \varepsilon}\right) \frac{dt}{\sin t}\).

The best solution was “again” submitted by Seong-Gu Jeong (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2009-22.

Evaluate the following limit:

\(\displaystyle \lim_{\varepsilon\to 0}\int_0^{2\varepsilon} \log\left(\frac{|\sin t-\varepsilon|}{\sin \varepsilon}\right) \frac{dt}{\sin t}\).

Let A=(a

_{ij}) be an n×n matrix such that a_{ij}=cos(i-j)θ and θ=2π/n. Determine the rank and eigenvalues of A.

The best solution was submitted by Seong-Gu Jeong (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2009-21.

Let A=(a_{ij}) be an n×n matrix such that a_{ij}=cos(i-j)θ and θ=2π/n. Determine the rank and eigenvalues of A.

Let e

_{n}be the expect value of the product x_{1}x_{2}…x_{n}where x_{1}is chosen uniformly at random in (0,1) and x_{k}is chosen uniformly at random in (x_{k-1},1) for k=2,3,…,n. Prove that \(\displaystyle \lim_{n\to \infty} e_n=\frac1e\).

The best solution was submitted by Seong-Gu Jeong (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2009-20.

Let e_{n} be the expect value of the product x_{1}x_{2} …x_{n} where x_{1} is chosen uniformly at random in (0,1) and x_{k} is chosen uniformly at random in (x_{k-1},1) for k=2,3,…,n. Prove that \(\displaystyle \lim_{n\to \infty} e_n=\frac1e\).

Let A and B be n×n matrices over the real field R. Prove that if A+B is invertible, then A(A+B)

^{-1}B=B(A+B)^{-1}A.

The best solution was submitted by SeungKyun Park (박승균), 2008학번. Congratulations!

Here is his Solution of Problem 2009-19.

Alternative solutions were submitted by 옥성민 (수리과학과 2003학번, +3), 노호성 (물리학과 2008학번, +3), 송지용 (수리과학과 2006학번, +3), 김현 (2008학번, +3), 정성구 (수리과학과 2007학번, +3), 이재송 (전산학과 2005학번, +3), 정지수 (수리과학과 2007학번, +3), 김호진 (2009학번, +3), 최석웅 (수리과학과 2006학번, +3), 김환문 (물리학과 2008학번, +3), 류종하 (서울대학교 전기과 2008학번). One incorrect solution was received. Thank you for the participation.