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=(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.
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.
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<…<ak<n be a sequence of integers such that gcd(ai,aj)=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.
Let C be a continuous and self-avoiding curve on the plane. A curve from A to B is called a C-segment if it connects points A and B, and is similar to C. A set S of points one the plane is called C-convex if for any two distinct points P and Q in S, all the points on the C-segment from P to Q is contained in S.
Prove that a bounded C-convex set with at least two points exists if and only if C is a line segment PQ.
(We say that two curves are similar if one is obtainable from the other by rotating, magnifying and translating.)
The best solution was submitted by Jae-song Lee (이재송), 전산학과 2005학번. Congratulations!
Here is his Solution of Problem 2009-14.
Alternative solutions were submitted by 최범준 (수리과학과 2007학번, +3), 정성구 (수리과학과 2007학번, +3). One incorrect solution was received.