# 2009-23 Irrational number

Prove that  $$\sqrt{2}+\sqrt{5}$$ is irrational.

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# Solution: 2009-22 Integral and Limit

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.

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# 2009-22 Integral and Limit

Evaluate the following limit:
$$\displaystyle \lim_{\varepsilon\to 0}\int_0^{2\varepsilon} \log\left(\frac{|\sin t-\varepsilon|}{\sin \varepsilon}\right) \frac{dt}{\sin t}$$.

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# Solution: 2009-21 Rank and Eigenvalues

Let A=(aij) be an n×n matrix such that aij=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.

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# 2009-21 Rank and Eigenvalues

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

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# Solution: 2009-20 Expectation

Let en be the expect value of the product x1x2 …xn where x1 is chosen uniformly at random in (0,1) and xk is chosen uniformly at random in (xk-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.

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# 2009-20 Expectation

Let en be the expect value of the product x1x2 …xn where x1 is chosen uniformly at random in (0,1) and xk is chosen uniformly at random in (xk-1,1) for k=2,3,…,n. Prove that $$\displaystyle \lim_{n\to \infty} e_n=\frac1e$$.

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