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.

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.

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!

Let A, B be \(3\times 3\) integer matrices such that A, A+B, A+2B, A+3B, A-B, A-2B, A-3B are invertible and their inverse matrices are all integer matrices. 

Prove that A+4B also has an inverse, and its inverse is again an integer matrix.

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

Here is his Solution of Problem 2008-3.