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 \( \mathbb{R}^n \).)

The best solution was submitted by 어수강. Congratulations!

Alternative solutions were submitted by 이종원 (+3), 장기정 (+3), 정성진 (+3), 채석주 (+1), 황성호 (+1). Thank you for your participation.

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.

The best solution was submitted by 황성호. Congratulations!

Alternative solutions were submitted by 정성진(+3), 이영민(+3), 채석주(+3), 이종원(+3), 장기정(+3), 배형진(+3), 남재현(+2), 김경민(+2), 박경호(+2), 서웅찬(+2). Thank you for your participation.