# Solution: 2015-17 Inverse of a minor

Let $$H$$ be an $$N \times N$$ positive definite matrix and $$G = H^{-1}$$. Let $$H’$$ be an $$(N-1) \times (N-1)$$ matrix obtained by removing the $$N$$-th row and the column of $$H$$, i.e., $$H’_{ij} = H_{ij}$$ for any $$i, j = 1, 2, \cdots, N-1$$. Let $$G’ = (H’)^{-1}$$. Prove that
$G_{ij} – G’_{ij} = \frac{G_{iN} G_{Nj}}{G_{NN}}$
for any $$i, j = 1, 2, \cdots, N-1$$.

The best solution was submitted by Park, Hun Min (박훈민, 수리과학과 2013학번). Congratulations!

Here is his solution of problem_2015_17.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 최인혁 (2015학번, +3), 박성혁 (수리과학과 2014학번, +3, solution), 신준형 (2015학번, +3), 이영민 (수리과학과 2012학번, +3), 장기정 (수리과학과 2014학번, +3), 함도규 (2015학번, +3).

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# Solution: 2015-2 Monochromatic triangle

Let $$T$$ be a triangle. Prove that if every point of a plane is colored by Red, Blue, or Green, then there is a triangle similar to $$T$$ such that all vertices of this triangle have the same color.

The best solution was submitted by 박훈민 (수리과학과 2013학번). Congratulations!

Here is his solution of problem 2015-2.

Alternative solutions were submitted by 국윤범/고경훈 (2015학번, +3 jointly / +2 each), 김경석 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 엄태현 (수리과학과 2012학번), 오동우 (2015학번, +3), 이명재 (수리과학과 2012학번, +3), 이수철 (2012학번, +2), 이영민 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 장기정 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 진우영 (수리과학과 2012학번, +3), 최인혁 (2015학번, +3). There was 1 incorrect solution (SML).

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# Concluding 2014 Fall

Thanks all for participating POW actively. Here’s the list of winners:

• 1st prize (Gold): Park, Minjae (박민재) – 수리과학과 2011학번
• 2nd prize (Silver): Chae, Seok Joo (채석주) – 수리과학과 2013학번
• 3rd prize (Bronze): Lee, Byeonghak (이병학) – 수리과학과 2013학번
• 4th prize: Park, Jimin (박지민) – 전산학과 2012학번
• 5th prize: Park, Hun Min (박훈민) – 수리과학과 2013학번

박민재 (2011학번) 30
채석주 (2013학번) 22
이병학 (2013학번) 20
박지민 (2012학번) 19
박훈민 (2013학번) 15
장기정 (2014학번) 14
허원영 (2014학번) 4
정성진 (2013학번) 3
김태겸 (2013학번) 3
윤준기 (2014학번) 3

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# Solution: 2014-22 Limit

For a nonnegative real number $$x$$, let $f_n(x)=\frac{\prod_{k=1}^{n-1} ((x+k)(x+k+1))}{ (n!)^2}$ for a positive integer $$n$$. Determine  $$\lim_{n\to\infty}f_n(x)$$.

The best solution was submitted by Hun-Min Park (박훈민), 수리과학과 2013학번. Congratulations!

Here is his solution of problem 2014-22.

Alternative solutions were submitted by 박민재 (수리과학과 2011학번, +3, his solution), 박지민 (전산학과 2012학번, +3), 이병학 (수리과학과 2013학번, +3), 채석주 (수리과학과 2013학번, +3).

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# Solution: 2014-03 Subadditive function

Let $$f: [0, \infty) \to \mathbb{R}$$ be a function satisfying the following conditions:

(1) For any $$x, y \geq 0$$, $$f(x+y) \geq f(x)+f(y)$$.

(2) For any $$x \in [0, 2]$$, $$f(x) \geq x^2 – x$$.

Prove that, for any positive integer $$M$$ and positive reals $$n_1, n_2, \cdots, n_M$$ with $$n_1 + n_2 + \cdots + n_M = M$$, we have

$f(n_1) + f(n_2) + \cdots + f(n_M) \geq 0.$

The best solution was submitted by 박훈민. Congratulations!

Alternative solutions were submitted by 김경석 (+3), 김일희 (+3), 안현수 (+3), 어수강 (+3), 윤성철 (+3), 이영민 (+3), 이종원 (+3), 장경석 (+3), 장기정 (+3), 정성진 (+3), 정진야 (+3), 조준영 (+3), 채석주 (+3), 황성호 (+3). Incorrect solutions were submitted by K.W.J., K.H.S., N.J.H, M.K.Y., S.W.C., L.H.B., C.W.H. (Some initials here might have been improperly chosen.)

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# Solution: 2013-18 Idempotent elements

Let $$R$$ be a ring of characteristic zero. Assume further that $$na \neq 0$$ for a positive integer $$n$$ and $$a \in R$$ unless $$a = 0$$. Suppose that $$e, f, g \in R$$ are idempotent (with respect to the multiplication) and satisfy $$e + f + g = 0$$. Show that $$e = f = g = 0$$. (An element $$a$$ is idempotent if $$a^2 = a$$. )

The best solution was submitted by 박훈민. Congratulations!

Similar solutions are submitted by 김동현(+3), 김호진(+3), 도수일(+3), 박민재(+3), 정성진(+3), 진우영(+3). Thank you for your participation.

Remark: Special thanks to 김동현, who first reported that the condition `characteristic zero’ is insufficient for the problem.

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# Solution: 2013-06 Inequality on the unit interval

Let $$f : [0, 1] \to \mathbb{R}$$ be a continuously differentiable function with $$f(0) = 0$$ and $$0 < f'(x) \leq 1$$. Prove that $\left( \int_0^1 f(x) dx \right)^2 \geq \int_0^1 [f(x)]^3 dx.$

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

Other solutions were submitted by 라준현(08학번, +3), 김호진(09학번, +3), 서기원(09학번, +3), 김범수(10학번, +3), 황성호(13학번, +3), 홍혁표(13학번, +3), 김준(13학번, +3), 전한솔(고려대 13학번, +3), 이시우(POSTECH 13학번, +3), 한대진(신현여중 교사, +3). Thank you for your participation.

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