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.

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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.

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2010-9 No zeros far away

Let M>0 be a real number. Prove that there exists N so that if n>N, then all the roots of $$f_n(z)=1+\frac{1}{z}+\frac1{{2!}z^2}+\cdots+\frac{1}{n!z^n}$$ are in the disk |z|<M on the complex plane.

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Solution: 2010-6 Identity on Binomial Coefficients

Prove that $$\displaystyle \sum_{m=0}^n \sum_{i=0}^m \binom{n}{m} \binom{m}{i}^3=\sum_{m=0}^n \binom{2m}{m} \binom{n}{m}^2$$.

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

Here is his Solution of Problem 2010-6.

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2010-8 Monochromatic Box

Let k be a postivive integer. Let f(k) be the minimum number n such that no matter how we color the integer points in {(x,y,z): 0<x,y,z≤n} with k colors, there always exist 8 monochromatic points forming the vertices of a box whose sides are parallel to xy- or yz- or xz- plane. Determine f(k).

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Solution: 2010-7 Cardinality

Let S be the set of non-zero real numbers x such that there is exactly one 0-1 sequence {an} satisfying $$\displaystyle \sum_{n=1}^\infty a_n x^{-n}=1$$. Prove that there is a one-to-one function from the set of all real numbers to S.

The best solution was submitted by Jeong, Seong Gu (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2010-7.

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Let S be the set of non-zero real numbers x such that there is exactly one 0-1 sequence {an} satisfying $$\displaystyle \sum_{n=1}^\infty a_n x^{-n}=1$$. Prove that there is a one-to-one function from the set of all real numbers to S.