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 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?
(Problem updated on Sep. 26, 8AM: gcd(ai,aj)=1.)
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).
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.
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.
What is the value of the following infinite series?
\(\displaystyle\sum_{n=2}^\infty \sum_{m=1}^{n-1} \frac{(-1)^n}{mn}\)
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.
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.)
KAIST Problem of the Week will continue in the fall semester of 2009. The first problem of 2009 Fall will be posted online on Sep. 4, Friday 3PM.