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 ( more resources) often!
