# Solution: 2011-19 Irreducible polynomial

Find all n≥2 such that the polynomial xn-xn-1-xn-2-…-x-1 is irreducible over the rationals.

The best solution was submitted by Gee Won Suh (서기원), 수리과학과 2009학번. Congratulations!

Here is his Solution of Problem 2011-19.

One incorrect solution by W.J. Kim was submitted.

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# Solution: 2011-15 Two matrices

Let n be a positive integer. Let ω=cos(2π/n)+i sin(2π/n). Suppose that A, B are two complex square matrices such that AB=ω BA. Prove that (A+B)n=An+Bn.

The best solution was submitted by Gee Won Suh (서기원), 수리과학과 2009학번. Congratulations!

Here is his Solution of Problem 2011-15.

Alternative solutions were submitted by 박민재 (2011학번, +3, alternative solution), 장경석 (2011학번, +3).

GD Star Rating # Concluding 2011 Spring

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

1st prize: Park, Minjae (박민재) – 2011학번

2nd prize: Kang, Dongyub (강동엽) – 전산학과 2009학번

3rd prize: Suh, Gee Won (서기원) – 수리과학과 2009학번
3rd prize: Lee, Jaeseok (이재석) – 수리과학과 2007학번

Congratulations!

In addition to these three people, I selected one more student to receive one notebook.

Kim, Ji Won (김지원) -수리과학과 2010학번 박민재 (2011학번) 31pts
강동엽 (2009학번) 24pts
서기원 (2009학번) 16pts
이재석 (2007학번) 16pts
김지원 (2010학번) 12pts
김치헌 (2006학번) 5pts
김인환 (2010학번) 3pts
김태호 (2011학번) 3pts
양해훈 (2008학번) 3pts
이동민 (2009학번) 2pts

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# Solution: 2011-4 A polynomial with distinct real zeros

Let n>2. Let f (x) be a degree-n polynomial with real coefficients. If f (x) has n distinct real zeros r1<r2<…<rn, then Rolle’s theorem implies that the largest real zero q of (x) is between rn-1 and rn. Prove that q>(rn-1+rn)/2.

The best solution was submitted by Gee Won Suh (서기원), 2009학번. Congratulations!

Here is his Solution of Problem 2011-4.

Alternative solutions were submitted by 박민재 (2011학번, +3), 강동엽 (전산학과 2009학번, +3), 김태호 (2011학번, +3), 김지원 (2010학번, +3), 이재석 (수리과학과 2007학번, +3), 김현수 (한국과학영재학교 3학년, +3), 구도완 (해운대고등학교 3학년, +3).

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# Solution: 2010-14 Combinatorial Identity

Let n be a positive integer. Prove that

$$\displaystyle \sum_{k=0}^n (-1)^k \binom{2n+2k}{n+k} \binom{n+k}{2k}=(-4)^n$$.

The best solution was submitted by Gee Won Suh (서기원), 2009학번. Congratulations!

Here is his Solution of Problem 2010-14.

Alternative solutions were submitted by 김치헌 (수리과학과 2006학번, +3), 정진명 (수리과학과 2007학번, +3), 박민재 (KSA-한국과학영재학교, +3), 오성진 (Princeton Univ.), Abhishek Verma (GET-SKEC NDEC, New Delhi).

Here are some interesting solutions.

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# Solution: 2010-11 Integral Equation

Let z be a real number. Find all solutions of the following integral equation: $$f(x)=e^x+z \int_0^1 e^{x-y} f(y)\,dy$$ for 0≤x≤1.

The best solution was submitted by Gee Won Suh (서기원), 2009학번. Congratulations!

Here is his Solution of Problem 2010-11.

Alternative solutions were submitted by 최홍석 (화학과 2006학번, +3), 정성구 (수리과학과 2007학번, +3).

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# Solution: 2010-10 Metric space of matrices

Let  Mn×n be the space of real n×n matrices, regarded as a metric space with the distance function

$$\displaystyle d(A,B)=\sum_{i,j} |a_{ij}-b_{ij}|$$

for A=(aij) and B=(bij).
Prove that $$\{A\in M_{n\times n}: A^m=0 \text{ for some positive integer }m\}$$ is a closed set.

The best solution was submitted by Gee Won Suh (서기원), 2009학번. Congratulations!

Here is his Solution of Problem 2010-10.

Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), 김치헌 (수리과학과 2006학번, +3), 강동엽 (2009학번, +2).

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