# Solution: 2014-13 Unit vectors

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.

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# Solution: 2013-09 Inequality for a sequence

Let $$N > 1000$$ be an integer. Define a sequence $$A_n$$ by
$A_0 = 1, \, A_1 = 0, \, A_{2k+1} = \frac{2k}{2k+1} A_{2k} + \frac{1}{2k+1} A_{2k-1}, \, A_{2k} = \frac{2k-1}{2k} \frac{A_{2k-1}}{N} + \frac{1}{2k} A_{2k-2}.$
Show that the following inequality holds for any integer $$k$$ with $$1 \leq k \leq (1/2) N^{1/3}$$.
$A_{2k-2} \leq \frac{1}{\sqrt{(2k-2)!}}.$

The best solution was submitted by 어수강, 서울대학교 석사과정. Congratulations!

An alternative solution was submitted by 라준현(08학번, +3). Thank you for your participation.

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