Favorite Math Competition Problems I'm running a weekly math contest for a summer camp and would like to compile a list of interesting problems. The problems may presuppose mathematical knowledge up to but not including Calculus.
For my purposes, a good problem is one that emphasizes cleverness over knowledge. For example, "Find the area of this triangle given some information" is a bad problem, since it's solution is likely just a matter of knowing lots of different facts about triangles.
 A: Here is a list of interesting problem sites.
A: Qbyte, MathsChallenge, and Gurmeet have a good deal of problems for the high school level. You'll probably want to check out some problem books too; Arthur Engel's Problem-Solving Strategies contains a wide range of problems, and Mathematical Olympiad Treasures by Andreescue and Enescu as well. There are also Mathematical Magazines dedicated to problems and problem solving, one such magazine is Crux Mathematicorum, and it also includes an "Olympiad Corner" for contest problems.
A: The Australian Mathematical Society Gazette http://www.austms.org.au/Gazette+-+past+issues has been running a Puzzle Corner for the last few years. Many of the puzzles rely on cleverness rather than advanced knowledge. 
Peter Winkler has published two excellent books of puzzles. http://www.math.dartmouth.edu/~pw/ is his home page. 
A: http://www.artofproblemsolving.com
A: Brilliant.org is also a new and running amazing source of problems in several subjects $-$ it is interactive and useful.
A: Given the two parabolas y = x^2 and y = -x^2, and that the upper parabola rolls without slipping on the lower parabola first one way and then the other, find the locus of the focus of the moving (ie upper) parabola.
From the 1974 Putnum competition (wording from memory).
A: The Tournament of the Towns problems are composed specifically to require nothing beyond high-school mathematics.  This wouldn't necessarily rule out problems that require calculus, but very few, if any, of the ones I can recall seeing have done so. 
The problems are graded in difficulty from moderate (3 points or fewer) to severe (approx. 6 points or more), and many of them are quite pretty.
