# Solution: 2013-01 Inequality involving eigenvalues and traces

Let $$A, B$$ be $$N \times N$$ symmetric matrices with eigenvalues $$\lambda_1^A \leq \lambda_2^A \leq \cdots \leq \lambda_N^A$$ and $$\lambda_1^B \leq \lambda_2^B \leq \cdots \leq \lambda_N^B$$. Prove that
$\sum_{i=1}^N |\lambda_i^A – \lambda_i^B|^2 \leq Tr (A-B)^2$

The best solution was submitted by 라준현, 08학번. Congratulations!

Alternative solutions were submitted by 김호진(09학번, +3), 서기원(09학번, +3), 곽걸담(11학번, +3), 김정민(12학번, +2), 홍혁표(13학번, +2). Thank you for your participation.

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# 2013-02 Functional equation

Let $$\mathbb{Z}^+$$ be the set of positive integers. Suppose that $$f : \mathbb{Z}^+ \to \mathbb{Z}^+$$ satisfies the following conditions.

i) $$f(f(x)) = 5x$$.

ii) If $$m \geq n$$, then $$f(m) \geq f(n)$$.

iii) $$f(1) \neq 2$$.

Find $$f(256)$$.

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