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.

 

Byoung Chan Lee (이병찬)

Prove that if x is a real number such that \(0<x\le \frac12\), then x can be represented as an infinite sum 

\(\displaystyle x=\sum_{k=1}^\infty \frac{1}{n_k}\),

where each \(n_k\) is an integer such that \(\frac{n_{k+1}}{n_k}\in \{3,4,5,6,8,9\}\).

The best solution was submitted by Byoung Chan Lee (이병찬), 수리과학과 2007학번. Congratulations! 

Click here for his Solution of Problem 2008-2.

Chiheon Kim (김치헌)

Let \(n\) be a positive integer. Let \(a_1,a_2,\ldots,a_k\) be distinct integers larger than \(n^{n-1}\) such that \(|a_i-a_j|<n\) for all \(i,j\).

Prove that the number of primes dividing \(a_1a_2\cdots a_k\) is at least \(k\).

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

Click here for his solution of Problem 2008-1.

This problem is equivalent to a theorem of Grimm (see his paper, A Conjecture on Consecutive Composite Numbers, The American Mathematical Monthly, Vol. 76, No. 10 (Dec., 1969), pp. 1126-1128). He conjectured that the same thing can be done without the lower bound \(n^{n-1}\). Laishram and Shorey verified Grimm’s conjecture when \(n<19000000000\).