# Solution: 2009-17 Relatively prime sequence

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).

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# 2009-17 Relatively prime sequence

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.)

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# Solution: 2009-16 Commutative ring

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).

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# 2009-16 Commutative ring

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.

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# Solution: 2009-15 Double Sum

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.

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# 2009-15 Double sum

What is the value of the following infinite series?

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

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# Solution: 2009-14 New notion on the convexity

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.

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# 2009-14 New notion on the convexity

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.)

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