# 2013-04 Largest eigenvalue of a symmetric matrix

Let $$H$$ be an $$N \times N$$ real symmetric matrix. Suppose that $$|H_{kk}| < 1$$ for $$1 \leq k \leq N$$. Prove that, if $$|H_{ij}| > 4$$ for some $$i, j$$, then the largest eigenvalue of $$H$$ is larger than $$3$$.

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# Solution: 2013-03 Hyperbolic cosine

Let $$t$$ be a positive real number and $$m$$ be a positive integer. Show that if both $$\cosh \, mt$$ and $$\cosh \, (m+1)t$$ are rational then $$\cosh \, t$$ is also rational.

The best solution was submitted by 홍혁표, 13학번. Congratulations!

Other solutions were submitted by 라준현(08학번, +3), 서기원(09학번, +3), 김호진(09학번, +3), 김범수(10학번, +3), 박지민(12학번, +3), 김정민(12학번, +2), 양지훈(10학번, +2), 황성호(13학번, +2). Thank you for your participation.

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# 2013-03 Hyperbolic cosine

Let $$t$$ be a positive real number and $$m$$ be a positive integer. Show that if both $$\cosh \, mt$$ and $$\cosh \, (m+1)t$$ are rational then $$\cosh \, t$$ is also rational.

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# Solution: 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 $$f(256)$$.

The best solution was submitted by 김호진, 09학번. Congratulations!

Similar solutions were also submitted by 황성호(13학번, +3), 양지훈(10학번, +3), 홍혁표(13학번, +3), 김준(13학번, +3), 서기원(09학번, +3), 이주호(12학번, +3), 박훈민(13학번, +3), 송유신(10학번, +3), 임현진(10학번, +3), 라준현(08학번, +3), 김정민(12학번, +3), 박지민(12학번, +3), 김태호(11학번, +3), 김범수(10학번, +3), 전한솔(고려대 13학번, +3), 어수강(서울대 석사과정, +3), 이시우(POSTECH 13학번, +3), 정우석(서강대 11학번, +3), 윤성철(홍익대 09학번, +3), 김재호(하나고, +3), 이정준(08학번, +2). Thank you for your participation.

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# 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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# 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

$\sum_{i=1}^N |\lambda_i^A – \lambda_i^B|^2 \leq Tr (A-B)^2$

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