Prove that, for any sequences of real numbers \( \{ a_n \} \) and \( \{ b_n \} \), we have
\[
\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{a_m b_n}{m+n} \leq \pi \left( \sum_{m=1}^{\infty} a_m^2 \right)^{1/2} \left( \sum_{n=1}^{\infty} b_n^2 \right)^{1/2}
\]
For integer \( n \geq 1 \), define
\[
a_n = \sum_{k=0}^{\infty} \frac{k^n}{k!}, \quad b_n = \sum_{k=0}^{\infty} (-1)^k \frac{k^n}{k!}.
\]
Prove that \( a_n b_n \) is an integer.
Let \(a\), \(b\) be distinct positive integers. Prove that there exists a prime \(p\) such that when dividing both \(a\) and \(b\) by \(p\), the remainder of \(a\) is less than the remainder of \(b\).
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.
Suppose that \( a_1, a_2, \cdots \) are positive real numbers. Prove that
\[
\sum_{n=1}^{\infty} (a_1 a_2 \cdots a_n)^{1/n} \leq e \sum_{n=1}^{\infty} a_n \,.
\]
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).
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 \(A\).)
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.
Let \( f: [0, \infty) \to \mathbb{R} \) be a function satisfying the following conditions:
(1) For any \( x, y \geq 0 \), \( f(x+y) \geq f(x)+f(y) \).
(2) For any \( x \in [0, 2] \), \( f(x) \geq x^2 – x \).
Prove that, for any positive integer \( M \) and positive reals \( n_1, n_2, \cdots, n_M \) with \( n_1 + n_2 + \cdots + n_M = M \), we have
\[ f(n_1) + f(n_2) + \cdots + f(n_M) \geq 0. \]
Determine all positive integers \(\ell\) such that \[ \sum_{n=1}^\infty \frac{n^3}{(n+1)(n+2)(n+3)\cdots (n+\ell)}\] converges and if it converges, then compute its value.