# 2014-07 Subsequence

Let $$a_1,a_2,\ldots$$ be an infinite sequence of positive real numbers such that $$\sum_{n=1}^\infty a_n$$ converges. Prove that for every positive constant $$c$$, there exists an infinite sequence $$i_1<i_2<i_3<\cdots$$ of positive integers such that $$| i_n-cn^3| =O(n^2)$$ and  $$\sum_{n=1}^\infty \left( a_{i_n} (a_1^{1/3}+a_2^{1/3}+\cdots+a_{i_n}^{1/3})\right)$$ converges.

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# Midterm break

The problem of the week will take a break during the midterm exam period and return on April 26, Friday. Good luck on your midterm exams!

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# Solution: 2014-06 Inequality with e

Suppose that $$a_1, a_2, \cdots$$ are positive real numbers. Prove that
$\sum_{n=1}^{\infty} (a_1 a_2 \cdots a_n)^{1/n} \leq e \sum_{n=1}^{\infty} a_n \,.$

The best solution was submitted by 정성진. Congratulations!

Alternative solutions were submitted by 김경석 (+3), 이영민 (+3), 이종원 (+3), 장기정 (+3), 정성진 (+3), 조준영 (+3), 황성호 (+2). Incorrect solutions were submitted by K.S.J., L.S.C. (Some initials here might have been improperly chosen.)

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# 2014-06 Inequality with e

Suppose that $$a_1, a_2, \cdots$$ are positive real numbers. Prove that
$\sum_{n=1}^{\infty} (a_1 a_2 \cdots a_n)^{1/n} \leq e \sum_{n=1}^{\infty} a_n \,.$

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# Solution: 2014-05 Nonnegative determinant

Let $$n$$, $$k$$ be positive integers and let $$A_1,A_2,\ldots,A_n$$ be $$k\times k$$ real matrices. Prove or disprove that $\det\left(\sum_{i=1}^n A_i^t A_i\right)\ge 0.$  (Here, $$A^t$$ denotes the transpose of the matrix $$A$$.)

The best (most elementary) solution was submitted by 김정민. Congratulations!

Alternative solutions were submitted by 조준영 (+3), 채석주 (+3), 이영민 (+3), 심병수 (+3), 박훈민 (+3), 장기정 (+3), 정성진 (+3), 황성호 (+3), 이종원 (+3), 김일희 (+2), 남재현 (+3), 박경호 (+3).

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# 2014-05 Nonnegative determinant

Let $$n$$, $$k$$ be positive integers and let $$A_1,A_2,\ldots,A_n$$ be $$k\times k$$ real matrices. Prove or disprove that $\det\left(\sum_{i=1}^n A_i^t A_i\right)\ge 0.$  (Here, $$A^t$$ denotes the transpose of the matrix $$A$$.)

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# Solution: 2014-04 Integer pairs

Prove that there exist infinitely many pairs of positive integers $$(m, n)$$ satisfying the following properties:

(1) gcd$$(m, n) = 1$$.

(2) $$(x+m)^3 = nx$$ has three distinct integer solutions.

The best solution was submitted by 이종원. Congratulations!

Alternative solutions were submitted by 김경석 (+3), 김은혜 (+3), 김일희 (+3), 김찬민 (+3), 박훈민 (+3), 안현수 (+3), 어수강 (+3), 윤성철 (+3), 이영민 (+3), 장경석 (+3), 장기정 (+3), 정성진 (+3), 조준영 (+3), 채석주 (+3), 황성호 (+3), 박경호 (+2), 조남경 (+2). An incorrect solutions was submitted by N.J.H. (Some initials here might have been improperly chosen.)

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