# 2011-16 Odd Sets with Even Intersection

Let A1, A2, A3, …, An be finite sets such that |Ai| is odd for all 1≤i≤n and |Ai∩Aj| is even for all 1≤i<j≤n. Prove that it is possible to pick one element ai in each set Ai so that a1, a2, …,an are distinct.

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# Solution: 2011-15 Two matrices

Let n be a positive integer. Let ω=cos(2π/n)+i sin(2π/n). Suppose that A, B are two complex square matrices such that AB=ω BA. Prove that (A+B)n=An+Bn.

The best solution was submitted by Gee Won Suh (서기원), 수리과학과 2009학번. Congratulations!

Here is his Solution of Problem 2011-15.

Alternative solutions were submitted by 박민재 (2011학번, +3, alternative solution), 장경석 (2011학번, +3).

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# 2011-15 Two matrices

Let n be a positive integer. Let ω=cos(2π/n)+i sin(2π/n). Suppose that A, B are two complex square matrices such that AB=ω BA. Prove that (A+B)n=An+Bn.

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# Solution: 2011-14 Invertible matrices

For a positive integer n>1, let f(n) be the largest real number such that for every n×n diagonal matrix M with positive diagonal entries, if tr(M)<f(n), then M-J is invertible. Determine f(n). (The matrix J is the square matrix with all entries 1.)

The best solution was submitted by Kyoungseok Jang(장경석), 2011학번. Congratulations!

Here is his Solution of Problem 2011-14.

Alternative solutions were submitted by 곽영진 (2011학번, +3), 박민재 (2011학번, +3), 라준현 (수리과학과 2008학번, +3), 서기원 (수리과학과 2009학번, +3), 배다슬 (수리과학과 2008학번, +3), 김범수 (수리과학과 2010학번, +3), 어수강 (서울대학교 수리과학부 대학원, +3), 조위지 (Stanford Univ. 물리학과 박사과정, +3).

PS. There were solutions without computing the determinant. Here is a Solution of Problem 2011-14 by 김범수.

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# 2011-14 Invertible matrices

For a positive integer n>1, let f(n) be the largest real number such that for every n×n diagonal matrix M with positive diagonal entries, if tr(M)<f(n), then M-J is invertible. Determine f(n). (The matrix J is the square matrix with all entries 1.)

(Due to a mistake, the problem is fixed at 3:30PM Friday.)

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# Solution: 2011-13 Sum of Partial Sums

Let a1, a2, … be a sequence of non-negative real numbers less than or equal to 1. Let $$S_n=\sum_{i=1}^n a_i$$ and $$T_n=\sum_{i=1}^n S_i$$. Prove or disprove that $$\sum_{n=1}^\infty a_n/T_n$$ converges. (Assume a1>0.)

The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!

Here is his Solution of Problem 2011-13. (There is a minor mistake in the proof.)

Alternative solutions were submitted by 어수강 (서울대학교 대학원, +2), 백진언 (한국과학영재학교, +2).

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# 2011-13 Sums of Partial Sums

Let a1, a2, … be a sequence of non-negative real numbers less than or equal to 1. Let $$S_n=\sum_{i=1}^n a_i$$ and $$T_n=\sum_{i=1}^n S_i$$. Prove or disprove that $$\sum_{n=1}^\infty a_n/T_n$$ converges. (Assume a1>0.)

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# Solution: 2011-12 Determinant

Let M=(mi,j)1≤i,j≤n be an n×n matrix such that mi,j=i(i+1)(i+2)…(i+j-2). (Note that m1,1=1.) What is the determinant of M?

The best solution was submitted by Seungkyun Park (박승균), 수리과학과 2008학번. Congratulations!

Here is his Solution of Problem 2011-12.

Alternative solutions were submitted by 조상흠 (수리과학과 2010학번, +3), 장경석 (2011학번, +3), 김원중 (2011학번, +3), 박민재 (2011학번, +3),   서기원 (수리과학과 2009학번, +3), 김범수 (2010학번, +3), 어수강 (서울대학교, +3),  조위지 (Stanford Univ. 물리학과 박사과정, +3).

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