Heres one:
There are 120 members in the Israeli parliament, and 40 committees. A committee can be assembled only when the number of committee members that are now present in the parliament hall, is odd (so that votes would be decisive). Is there a way to assign the parliament members to committees, so that in all times, except for when the hall is completely empty, there would be at least one committee that can be assembled?
Or this:
Let F be a finite field and c, d positive integers. Suppose for each x in Fc+d we have a real number v(x), such that for each c-dimensional affine subspace of Fc+d, the sum of v(x) over all the points in the subspace yields zero. Prove that v(x) equals 0 for all x in Fc+d.