Monthly Archives: October 2008

Solution: 2008-6 Many primes

Let f(x) be a polynomial with integer coefficients. Prove that if f(x) is not constant, then there are infinitely many primes p such that \(f(x)\equiv 0\pmod p\) has a solution x.

The best solution was submitted by Yang, Hae Hun  (양해훈), 2008학번. Congratulations!

Here is his Solution of Problem 2008-6.

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Solution: 2008-5 Monochromatic lines

Suppose that P is a finite set of points in the plane colored by red or blue. Show that if no straight line contains all points of P, then there exists a straight line L with at least two points of P on L such that all points on \(P\cap L\) have the same color.

The best solution was submitted by Haewon Yoon (윤혜원), 수리과학과 2004학번. Congratulations!

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Solution: 2008-4 Limit

 

Let \(a_1=\sqrt{1+2}\),
\(a_2=\sqrt{1+2\sqrt{1+3}}\),
\(a_3=\sqrt{1+2\sqrt{1+3\sqrt{1+4}}}\), …,
\(a_n=\sqrt{1+2\sqrt{1+3\sqrt {\cdots \sqrt{\sqrt{\sqrt{\cdots\sqrt{1+n\sqrt{1+(n+1)}}}}}}}}\), … .

Prove that \(\displaystyle\lim_{n\to \infty} \frac{a_{n+1}-a_{n}}{a_n-a_{n-1}}=\frac12\).

The best solution was submitted by Jaehoon Kim (김재훈), 수리과학과 2003학번. Congratulations! 

Here is his Solution of Problem 2008-4.

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2008-5 Monochromatic lines (10/2)

Suppose that P is a finite set of points in the plane colored by red or blue. Show that if no straight line contains all points of P, then there exists a straight line L with at least two points of P on L such that all points on \(P\cap L\) have the same color.

P는 평면 상에 점들의 집합으로, 각 점은 파랑색 혹은 빨간색으로 칠해져있다. 모든 P의 점을 포함하는 직선이 없다면, P의 두 점이상을 포함하는 직선 중에 \(P\cap L\)의 모든 점이 같은 색이 되도록 하는 직선이 있음을 보여라.

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