User:Temperal/The Problem Solver's Resource Tips and Tricks
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Other Tips and Tricks
This is a collection of general techniques for solving problems.
- Don't be afraid to use casework! Sometimes it's the only way. (But be VERY afraid to use brute force.)
- Remember that substitution is a useful technique! Example problem:
Example Problem 1
If and , find .
Solution
Let , . Thus, , , so , hence , which turns out to be .
This technique can also be used to solve quadratics of high degrees, i.e. ; let , and solve from there.
- Remember the special properties of odd numbers: For any odd number , for some integer , and for some positive integer .
Example Problem 2
How many quadruples are there such that and are all odd?
Solution
Since they're odd, can each be expressed as for some positive integer (or zero) . Thus:
Binomial coefficients will yield the answer of .
- The AM-GM and Trivial inequalities are more useful than you might imagine!
- Memorize, memorize, memorize the following things:
- The trigonometric facts.
- Everything on the Combinatorics page.
- Integrals and derivatives, especially integrals.
Remember, though, don't memorize without understanding!
- Test your skills on practice AIMEs (<url>resources.php more resources</url>) often!