The Art of Problem Solving (AopS) site encourages, teaches and promotes mathematics competitions from small local ones right up to the International Mathematical Olympiad (IMO). As it says on its front page:
Is math class too easy for you? Looking for a greater challenge?
You’ve come to the right place.
It has an excellent introduction to siteÂ and a forum to discuss problems. The forum has an RSS feed where students pose new problems every day. So if you’re “looking for a greater challenge” subscribe to this feed. Here is a random sample of some of the problems posed recently, which range from very easy to extremely difficult. Click on the problem number to go to the discussion on it.
- 1. We define addition in a different wayÂ to usual; an addition statement is true only if the letters in the addends is a rearrangement of the letters in the sum. For example,
10 + 6 = 16?
TEN + SIX = TENSIX = SIXTEN, but to be 16 it would need another E.
Find a “true” addition a + b = c + d.
2. Prove that .
3. Let be nonzero real numbers. Find all ordered pairs such that .
4. is a continuous complex-valued function satisfying:
5. If and are relatively coprime, find all possible values of .
6. Let be three angles ofÂ . Prove that .
7. For each function which is defined for all real numbers and satisfies and determine the value of .
8. Let be positive reals such that . Prove that .
9. For , we define the numbers . Find the last digit of the number .
10. The product of several distinct positive integers is divisible by . Determine the minimum value the sum of such numbers can take.