Difference between revisions of "User:Temperal/The Problem Solver's Resource Tips and Tricks"
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*Don't be afraid to use casework! Sometimes it's the only way. (But be VERY afraid to use brute force.) | *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: | *Remember that substitution is a useful technique! Example problem: | ||
− | ===Example Problem | + | ===Example Problem 1=== |
If <math>\tan x+\tan y=25</math> and <math>\cot x+\cot y=30</math>, find <math>\tan(x+y)</math>. | If <math>\tan x+\tan y=25</math> and <math>\cot x+\cot y=30</math>, find <math>\tan(x+y)</math>. | ||
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*Remember the special properties of odd numbers: For any odd number <math>o</math>, <math>o=2n\pm 1</math> for some integer <math>n</math>, and <math>o=a^2-(a-1)^2</math> for some positive integer <math>a</math>. | *Remember the special properties of odd numbers: For any odd number <math>o</math>, <math>o=2n\pm 1</math> for some integer <math>n</math>, and <math>o=a^2-(a-1)^2</math> for some positive integer <math>a</math>. | ||
− | ===Example Problem | + | ===Example Problem 2=== |
How many quadruples <math>(a,b,c,d)</math> are there such that <math>a+b+c+d=98</math> and <math>a,b,c,d</math> are all odd? | How many quadruples <math>(a,b,c,d)</math> are there such that <math>a+b+c+d=98</math> and <math>a,b,c,d</math> are all odd? | ||
Latest revision as of 20:06, 10 January 2009
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing the tips and tricks section. |
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!