# 2010-21 Limit

Let $$a_1=0$$, $$a_{2n+1}=a_{2n}=n-a_n$$. Prove that there exists k such that $$\lvert a_k- \frac{k}{3}\rvert >2010$$ and yet $$\lim_{n\to \infty} \frac{a_n}{n}=\frac13$$.

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# Solution: 2010-20 Monochromatic line

Let X be a finite set of points on the plane such that each point in X is colored with red or blue and there is no line having all points in X. Prove that there is a line L having at least two points of X such that all points in L∩X have the same color.

The best solution was submitted by Minjae Park (박민재), 한국과학영재학교 (KSA). Congratulations!

Here is his Solution of Problem 2010-20.

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# Solution:2010-19 Fixed Points

Suppose that $$V$$ is a vector space of dimension $$n>0$$ over a field of characterstic $$p\neq 0$$. Let $$A: V\to V$$ be an affine transformation. Prove that there exist $$u\in V$$ and $$1\le k\le np$$ such that $A^k u = u.$

The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-19.

An alternative solution was submitted by 박민재 (KSA-한국과학영재학교, +3).

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# 2010-20 Monochromatic line

Let X be a finite set of points on the plane such that each point in X is colored with red or blue and there is no line having all points in X. Prove that there is a line L having at least two points of X such that all points in L∩X have the same color.

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# 2010-19 Fixed point

Suppose that $$V$$ is a vector space of dimension $$n>0$$ over a field of characterstic $$p\neq 0$$. Let $$A: V\to V$$ be an affine transformation. Prove that there exist $$u\in V$$ and $$1\le k\le np$$ such that $A^k u = u.$

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# Solution: 2010-18 Limit of a differentiable function

Let f be a differentiable function. Prove that if $$\lim_{x\to\infty} (f(x)+f'(x))=1$$, then $$\lim_{x\to\infty} f(x)=1$$.

The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-18.

Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), 서기원 (수리과학과 2009학번, +3), 심규석 (수리과학과 2007학번, +3), 진우영 (KSA-한국과학영재학교, +3), 박민재 (KSA-한국과학영재학교, +2), 한대진 (?, +2), 문정원 (성균관대학교 수학교육과, +2).

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# 2010-18 Limit of a differentiable function

Let f be a differentiable function. Prove that if $$\lim_{x\to\infty} (f(x)+f'(x))=1$$, then $$\lim_{x\to\infty} f(x)=1$$.

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