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.

Suppose that we color integers 1, 2, 3, …, n with three colors so that each color is given to more than n/4 integers. Prove that there exist x, y, z such that x+y=z and x,y,z have distinct colors.

The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2009-12.

Alternative solutions were submitted by 백형렬 (수리과학과 2003학번, +3), 조강진 (2009학번, +2), 권상훈 (수리과학과 2006학번, +2).

Does there exist a set of circles on the plane such that every line intersects at least one but at most 100 of them?

The best solution was submitted by Hyung Ryul Baik (백형렬), 수리과학과 2003학번. Congratulations!

Here is his Solution of Problem 2009-11.

There were 2 other incorrect solutions submitted.

Reference: L. Yang, J. Zhang, and W. Zhang, On number of circles intersected by a line, J. Combin. Theory Ser. A, 98 (2002), pp. 395–405.

Find all real-valued continuous function f on the reals such that f(x)=f(cos x) for every real number x.