Let \(a_0=a\) and \(a_{n+1}=a_n (a_n^2-3)\). Find all real values \(a\) such that the sequence \(\{a_n\}\) converges.

 Let \(\mathcal F\) be a collection of subsets (of size r) of a finite set E such that \(X\cap Y\neq\emptyset\) for all \(X, Y\in \mathcal F\). Prove that there exists a subset S of E such that \(|S|\le (2r-1)\binom{2r-3}{r-1}\) and \(X\cap Y\cap S\neq\emptyset\) for all \(X,Y\in\mathcal F\).

The best solution was submitted by Hyung Ryul Baik (백형렬), 수리과학과 2003학번. Congratulations!

Click here for his Solution of Problem 2009-3.

Let \(a_1<\cdots\) be a sequence of positive integers such that \(\log a_1, \log a_2,\log a_3,\cdots\) are linearly independent over the rational field \(\mathbb Q\). Prove that \(\lim_{k\to \infty} a_k/k=\infty\).

The best solution was submitted by SangHoon Kwon (권상훈), 수리과학과 2006학번. Congratulations!

Click here for his Solution of Problem 2009-2.

There were 3 other submitted solutions which will earn points: 김치헌+3, 김린기+3,  조강진+2.

Chiheon Kim (김치헌)

Let \(a_1\le a_2\le \cdots \le a_n\) be integers. Prove that

\(\displaystyle\prod_{1\le j<i\le n} \frac{a_i-a_j}{i-j}\)

is an integer.

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

Click here for his Solution of Problem 2009-1.

Find all real numbers \(\lambda\) and the corresponding functions \(f\) such that the equation 
\(\displaystyle \int_0^1 \min(x,y) f(y) \,dy=\lambda f(x)\)
has a non-zero solution \(f\) that is continuous on the interval [0,1]. 

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

Here is his Solution of Problem 2008-12.

Let a, b, c, d be positive rational numbers. Prove that if \(\sqrt a+\sqrt b+\sqrt c+\sqrt d\) is rational, then each of \(\sqrt a\), \(\sqrt b\), \(\sqrt c\), and \(\sqrt d\) is rational.

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

Here is his Solution of Problem 2008-11.

Let \(x_1,x_2,\ldots,x_n\) be nonnegative real numbers. Show that
\(\displaystyle \left(\sum_{i=1}^n x_i\right) \left(\sum_{i=1}^n x_i^{n-1}\right) \le (n-1) \sum_{i=1}^n x_i^n + n \prod_{i=1}^n x_i \). 

The best solution was submitted by Sang Hoon Kwon (권상훈), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2008-10.

Let \(\mathbb{R}\) be the set of real numbers and let \(\mathbb{N}\) be the set of positive integers. Does there exist a function \(f:\mathbb{R}^3\to \mathbb{N}\) such that f(x,y,z)=f(y,z,w) implies x=y=z=w?

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