Patriotic chameleons

This is a problem based on one I saw on someone's Putnam Exam review sheet.

Puzzle

For next year's 4th of July party, you buy 2011 chameleons: some red, some white and some blue. These chameleons have the strange property that whenever two of different colors meet, they both change into the third color. Must there be a color among red, white and blue for which it is impossible for all of the chameleons to simultaneously become that color?

Help

Hint

You know you have some chameleons of each color. Do you know that there must be two colors with a different number of chameleons in each? What can that difference be? What happens to that difference when two chameleons meet?

Answer

Yes, there must always be such a color.

Solution

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