How many numbers between 1 and 1,000,000 are divisible by 2, 3, 4, 5, 6, 7, 8, 9, and 10?
(Originally published in the Mathematical Mayhem part of Crux Mathematicorum in um, March 2010, I think. Someone remind me to check that.)
And if that’s too easy for you: 201 people of 5 nationalities are at a party. In every group of 6 people, at least two people are the same age. Prove that there are at least five people of the same age, gender, and nationality.
(Originally given as a problem in the Spanish Matematics Olympiad. Need to check the year on that, too.)
[Edited: I should probably add something about the level of these problems. The first one is accessible to anyone who knows what divisibility is. The second one is hard…I’m still working on it, though if you paid more attention in combinatorics than I did, it’ll probably be easy for you.]