# Solution: 2009-5 Random points and the origin

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.

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# Solution: 2009-1 Integer or not

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.

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# Solution: 2008-1 Distinct primes

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 $$n<19000000000$$.

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