# 2008-8 Positive eigenvalues

Let A be a 0-1 square matrix. If all eigenvalues of A are real positive, then those eigenvalues are all equal to 1.

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# Solution: 2008-7 Find all real solutions

Find all real solutions of $$3^x + 5^{x^2} = 4^x + 4^{x^2}$$.

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

Here is his Solution of 2008-7.

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# 2008-7 Find all real solutions

Find all real solutions of $$3^x + 5^{x^2} = 4^x + 4^{x^2}$$.

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# 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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# No problem will be posted this week

No problem will be posted this week (Oct 16). Good luck with your midterm exam! Next problem will be posted on Oct 23.

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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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# 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.

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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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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.