# Solution: 2016-10 Factorization

Suppose that $$A$$ is an $$n \times n$$ matrix with integer entries and $$\det A = p_1^{e_1} p_2^{e_2} \dots p_k^{e_k}$$ for primes $$p_1, p_2, \dots, p_k$$ and positive integers $$e_1, e_2, \dots, e_k$$. Prove that there exist $$n \times n$$ matrices $$B_1, B_2, \dots, B_k$$ with integer entries such that $$A = B_1 B_2 \dots B_k$$ and $$\det B_1 = p_1^{e_1}, \det B_2 = p_2^{e_2}, \dots, \det B_k = p_k^{e_k}$$.

The best solution was submitted by Lee, Sangmin (이상민, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2016-10.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +2), 박정우 (한국과학영재학교 2016학번, +2).

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# 2016-11 Infinite series

For a positive integer $$n$$, define $$f(n)$$ by
$f(n) = \begin{cases} 0 & \text{ if } n \equiv 0 \pmod{5} \\ 1 & \text{ if } n \equiv \pm 1 \pmod{5} \\ -1 & \text{ if } n \equiv \pm 2 \pmod{5} \end{cases}.$
Compute the infinite series
$\sum_{n=1}^{\infty} \frac{f(n)}{n} = 1 – \frac{1}{2} – \frac{1}{3} + \frac{1}{4} + \frac{1}{6} – \dots.$

(This is the last problem of this semester. Thank you.)

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# Solution: 2016-9 Determinant of a matrix

Let $$A=(a_{ij})_{ij}$$ be an $$n\times n$$ matrix, where $a_{ij}=\begin{cases} 3 &\text{if }i=j,\\ (-1)^{\lvert i-j\rvert}&\text{otherwise.}\end{cases}$ Compute the determinant of $$A$$.

The best solution was submitted by Jang, Kijoung (장기정, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2016-9.

Alternative solutions were submitted by 강한필 (2016학번, +3), 국윤범 (수리과학과 2015학번, +3), 김태균 (2016학번, +3), 박정우 (한국과학영재학교 2016학번, +3), 송교범 (서대전고등학교 3학년, +3), 이상민 (수리과학과 2014학번, +3), 이원웅 (건국대 수학과 2014학번, +3), 최백규 (2016학번, +3), 유찬진 (수리과학과 2015학번, +2), 윤준기 (전기및전자공학부 2014학번, +2).

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# 2016-10 Factorization

Suppose that $$A$$ is an $$n \times n$$ matrix with integer entries and $$\det A = p_1^{e_1} p_2^{e_2} \dots p_k^{e_k}$$ for primes $$p_1, p_2, \dots, p_k$$ and positive integers $$e_1, e_2, \dots, e_k$$. Prove that there exist $$n \times n$$ matrices $$B_1, B_2, \dots, B_k$$ with integer entries such that $$A = B_1 B_2 \dots B_k$$ and $$\det B_1 = p_1^{e_1}, \det B_2 = p_2^{e_2}, \dots, \det B_k = p_k^{e_k}$$.

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# Solution: 2016-8 Limit

Compute $\lim_{n\to \infty} \cos^{2016} (\pi\sqrt{n^2+4n+9}).$

The best solution was submitted by Kang, Hanpil (강한필), 2016학번. Congratulations!

Here is his solution of problem 2016-8.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 이준호 (2016학번, +3), 장기정 (수리과학과 2014학번, +3), 유찬진 (수리과학과 2015학번, +3), 배형진 (마포고등학교 2학년, +3), 박기연 (2016학번, +3), 유현우 (한양대학교 화학공학과 2013학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 김재현 (2016학번, +3), 이예찬 (오송고등학교 교사, +3), 최백규 (2016학번, +3), 장창환 (기계공학과 2015학번, +3), 한대진 (인천예일중학교 교사, +3), 박은구 (연세대 수학과 대학원생, +3), 김태균 (2016학번, +3), 박정우 (한국과학영재학교 2016학번, +3), 이원웅 (건국대 수학과 2014학번, +3), 이상민 (수리과학과 2014학번, +3).

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# 2016-9 Determinant of a matrix

Let $$A=(a_{ij})_{ij}$$ be an $$n\times n$$ matrix, where $a_{ij}=\begin{cases} 3 &\text{if }i=j,\\ (-1)^{\lvert i-j\rvert}&\text{otherwise.}\end{cases}$ Compute the determinant of $$A$$.

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# Solution: 2016-7 Sum-free

For a set $$A \subset \mathbb{R}$$, let $$f(A)$$ be the size of the largest set $$B \subset A$$ such that $$(B+B) \cap B = \emptyset$$. For a positive integer $$n$$, let $$f(n) = \min_{0 \notin A, |A|=n} f(A)$$. Prove that $$f(n) \geq n/3$$.

The best solution was submitted by Kook, Yun Bum (국윤범, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-7.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 장기정 (수리과학과 2014학번, +3).

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# 2016-8 Limit

Compute $\lim_{n\to \infty} \cos^{2016} (\pi\sqrt{n^2+4n+9}).$

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# Solution: 2016-5 Partition into 4 sets

Let $$A_1,A_2,\ldots,A_n$$ be subsets of $$\{1,2,\ldots,n\}$$ such that $$i\notin A_i$$ for all $$i$$. Prove that there exist four sets $$C_1,C_2,C_3,C_4$$ such that $$C_1\cup C_2\cup C_3\cup C_4=\{1,2,\ldots,n\}$$ and for all $$i$$ and $$j$$, if $$i\in C_j$$, then $$\lvert A_i\cap C_j\rvert \le \frac12 \lvert A_i\rvert$$.

The best solution was submitted by Kook, Yun Bum (국윤범, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-5.

Alternative solutions were submitted by 이준호 (2016학번, +2), 김경석 (연세대학교 의예과 2016학번, +2). An incorrect solution was received.

Note: There is a simpler solution.

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