If we choose n points on a circle randomly and independently with uniform distribution, what is the probability that the center of the circle is contained in the interior of the convex hull of these n points?
The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!
There were 2 incorrect solutions submitted.
Click here for his Solution of Problem 2009-5.
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.
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 \(n<19000000000\).