Difference between revisions of "User:Temperal/The Problem Solver's Resource Tips and Tricks"
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(→Example Problem Number 1) |
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− | *Remember the special properties of odd numbers: For any odd number <math>o</math>, <math>o=2n | + | *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 Number 2=== | ||
+ | How many quadruples <math>(a,b,c,d)</math> such that <math>a+b+c+d=98</math> where <math>a,b,c,d</math> are all odd? | ||
+ | |||
+ | ====Solution==== | ||
+ | Since they're odd, <math>a, b, c, d</math> can each be expressed as <math>2n+1</math> for some positive integer (or zero) <math>n</math>. | ||
+ | Thus: | ||
+ | <math>2n_1-1+2n_2-1+2n_3+1+2n_4+1=98</math> | ||
+ | |||
+ | <math>\Rightarrow 2(n_1+n_2+n_3+n_4)+4=98</math> | ||
+ | |||
+ | <math>\Rightarrow 2(n_1+n_2+n_3+n_4)=94</math> | ||
+ | |||
+ | <math>\Rightarrow n_1+n_2+n_3+n_4=47[/latex] | ||
+ | Binomial coefficients will yield the answer of </math>\boxed{19600}$. | ||
+ | |||
+ | ---- | ||
*The AM-GM and Trivial inequalities are more useful than you might imagine! | *The AM-GM and Trivial inequalities are more useful than you might imagine! |
Revision as of 12:43, 13 October 2007
Other Tips and TricksThis is a collection of general techniques for solving problems.
Example Problem Number 1If SolutionLet This technique can also be used to solve quadratics of high degrees, i.e.
Example Problem Number 2How many quadruples SolutionSince they're odd,
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