Prove that finitely many squares on the plane with total area at least 3 can cover the unit square.

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

Here is his Solution of Problem 2010-1.

Alternative solutions were submitted by 임재원 & 서기원 (2009학번, +3 -> +2, +2 each) and 권용찬 (2009학번, +2; almost correct). Thank you for participation.

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

The best solution was submitted by Seong min Ok (옥성민), 수리과학과 2003학번. Congratulations!

Here is his Solution of Problem 2009-23.

Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), 심규석 (수리과학과 2007학번, +3), 이재송 (전산학과 2005학번).

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.

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 e_n be the expect value of the product x₁x₂ …x_n where x₁ 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 A and B be n×n matrices over the real field R. Prove that if A+B is invertible, then A(A+B)^-1B=B(A+B)^-1A.

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.

Suppose y(x)≥0 for all real x. Find all solutions of the differential equation \(\frac{dy}{dx}=\sqrt{y}\), y(0)=0.

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

Here is his Solution of Problem 2009-18.

There were 2 incorrect submissions.

Let 1≤a₁<a₂<…<a_k<n be a sequence of integers such that gcd(a_i,a_j)=1 for all 1≤i<j≤k. What is the maximum value of k?

The best solution was submitted by Yeon Sig Lyu (류연식), 2008학번. Congratulations!

Here is his Solution of Problem 2009-17.

Alternative solutions were submitted by Prach Siriviriyakul (2009학번, +3), 정성구 (수리과학과 2007학번, +3), 김치헌 (수리과학과 2006학번, +3), 옥성민 (수리과학과 2003학번, +3).

Let k>1 be a fixed integer. Let π be a fixed nonidentity permutation of {1,2,…,k}. Let I be an ideal of a ring R such that for any nonzero element a of R, aI≠0 and Ia≠0 hold.

Prove that if \(a_1 a_2\ldots a_k=a_{\pi(1)} a_{\pi(2)} \ldots a_{\pi(k)}\)  for any elements \(a_1, a_2,\ldots,a_k \in I\), then R is commutative.

The best solution was submitted by Yang, Hae Hun (양해훈), 2008학번. Congratulations!

Here is his Solution of Problem 2009-16.

An alternative solution was submitted by 정성구(수리과학과 2007학번, +3).

What is the value of the following infinite series?

\(\displaystyle\sum_{n=2}^\infty \sum_{m=1}^{n-1} \frac{(-1)^n}{mn}\)

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

Here is his Solution of Problem 2009-15.

An alternative solution was submitted by 김호진 (2009학번, +2). His alternative solution did not check whether the swapping two infinite sums can be done.