Splitting Points

From Five Hundred Mathematical Challenges:

Prove that if an even number of points are placed in a plane, it is always possible to draw a line such that half of the points are on one side of the line, and half the points are on the other side.

Dealing Cards

Mildly adapted from the 1895 Eötvös competition:

You have a deck of n cards, to be dealt to two players. The players do not have to receive the same number of cards. How many ways are there to do the dealing?

Some More Number Theory

From the 1901 Eötvös Contest:

Let n be a positive integer. Prove that 1ⁿ + 2ⁿ + 3ⁿ + 4ⁿ is divisible by 5 if and only if n is not divisible by 4.

Pirates

Here's a fun one from the Rice math competition General test in 2008:

Four pirates are dividing up 2008 gold pieces. They take turns, in order of rank, proposing ways to distribute the gold. If at least half the pirates agree to a proposal, it is enacted; otherwise, the proposer walks the plank. If no pirate ever agrees to a proposal that gives him nothing, how many gold pieces does the highest-ranking pirate end up with? (Assume all pirates are perfectly rational and act in self-interest, i.e. a pirate will never agree to a proposal if he knows he can gain more coins by rejecting
it.)

Also, consider how the answer might be generalized to arbitrary numbers of pirates and gold pieces.

Some Number Theory

Here's a question from last year's Math Prize For Girls competition:

When (&radic3 + 5)¹⁰³ - (&radic3 - 5)¹⁰³ is divided by 9, what is the remainder?