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 \(n<19000000000\).