Prove that, for any unit vectors \( v_1, v_2, \cdots, v_n \) in \( \mathbb{R}^n \), there exists a unit vector \( w \) in \( \mathbb{R}^n \) such that \( \langle w, v_i \rangle \leq n^{-1/2} \) for all \( i = 1, 2, \cdots, n \). (Here, \( \langle \cdot, \cdot \rangle \) is a usual scalar product in \( \mathbb{R}^n \).)
Category Archives: problem
2014-12 Rational ratios in a triangle
Determine all triangles ABC such that all of \( \frac{AB}{BC}, \frac{BC}{CA}, \frac{CA}{AB}, \frac{\angle A}{\angle B}, \frac{\angle B}{\angle C}, \frac{\angle C}{\angle A}\) are rational.
2014-11 Subsets of a countably infinite set
Prove or disprove that every uncountable collection of subsets of a countably infinite set must have two members whose intersection has at least 2014 elements.
2014-10 Inequality with pi
Prove that, for any sequences of real numbers \( \{ a_n \} \) and \( \{ b_n \} \), we have
\[
\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{a_m b_n}{m+n} \leq \pi \left( \sum_{m=1}^{\infty} a_m^2 \right)^{1/2} \left( \sum_{n=1}^{\infty} b_n^2 \right)^{1/2}
\]
2014-09 Product of series
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.
2014-08 Two positive integers
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 \(b\).
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.
2014-06 Inequality with e
Suppose that \( a_1, a_2, \cdots \) are positive real numbers. Prove that
\[
\sum_{n=1}^{\infty} (a_1 a_2 \cdots a_n)^{1/n} \leq e \sum_{n=1}^{\infty} a_n \,.
\]
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).
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 \(A\).)