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This one, by Dr. Titu Andreescu (of USAMO fame), is elementary in the sense that the solution to the problem doesn’t require anything more than arguments involving parity and congruences. I have the solution with me but I won’t post it on my blog until Jan 19, 2008, which is when the deadline for submission is. By the way, the problem (in the senior section) is from the issue of Mathematical Reflections, 2007.
Problem: Find the least odd positive integer such that for each prime is divisible by at least four (distinct) primes.