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Difference between revisions of "User:Temperal/The Problem Solver's Resource Tips and Tricks"

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Let <math>X = \tan x</math>, <math>Y = \tan y</math>. Thus, <math>X + Y = 25</math>, <math>\frac{1}{X} + \frac{1}{Y} = 30</math>, so <math>XY = \frac{5}{6}</math>, hence <math>\tan(x+y)=\frac{X+Y}{1-XY}</math>, which turns out to be <math>\boxed{150}</math>.
 
Let <math>X = \tan x</math>, <math>Y = \tan y</math>. Thus, <math>X + Y = 25</math>, <math>\frac{1}{X} + \frac{1}{Y} = 30</math>, so <math>XY = \frac{5}{6}</math>, hence <math>\tan(x+y)=\frac{X+Y}{1-XY}</math>, which turns out to be <math>\boxed{150}</math>.
  
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This technique can also be used to solve quadratics of high degrees, i.e. <math>x^16+x^4+6=0</math>; let <math>y=x^4</math>, and solve from there.
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This technique can also be used to solve quadratics of high degrees, i.e. <math>x^{16}+x^4+6=0</math>; let <math>y=x^4</math>, and solve from there.
  
 
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*Remember the special properties of odd numbers: For any odd number <math>o</math>, <math>o=2n-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-1</math> for some integer <math>n</math>, and <math>o=a^2-(a-1)^2</math> for some positive integer <math>a</math>.
  
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*The AMGM and Trivial inequalities are more useful than you might imagine!
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*The AM-GM and Trivial inequalities are more useful than you might imagine!
  
 
*Memorize, memorize, memorize the following things:
 
*Memorize, memorize, memorize the following things:
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#Integrals and derivatives, especially integrals.
 
#Integrals and derivatives, especially integrals.
  
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*Test your skills on [http://mathlinks.ro/Forum/resources.php practice AIMEs] often!  
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*Test your skills on practice [[AIME]]s (<url>resources.php more resources</url>) often!  
  
 
[[User:Temperal/The Problem Solver's Resource|Back to Introduction]]
 
[[User:Temperal/The Problem Solver's Resource|Back to Introduction]]
 
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Revision as of 20:15, 10 October 2007



The Problem Solver's Resource
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing Other Tips and Tricks.

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 Number 1

If $\tan x+\tan y=25$ and $\cot x+\ cot y=30$, find $\tan(x+y)$.

Solution

Let $X = \tan x$, $Y = \tan y$. Thus, $X + Y = 25$, $\frac{1}{X} + \frac{1}{Y} = 30$, so $XY = \frac{5}{6}$, hence $\tan(x+y)=\frac{X+Y}{1-XY}$, which turns out to be $\boxed{150}$.

This technique can also be used to solve quadratics of high degrees, i.e. $x^{16}+x^4+6=0$; let $y=x^4$, and solve from there.

  • Remember the special properties of odd numbers: For any odd number $o$, $o=2n-1$ for some integer $n$, and $o=a^2-(a-1)^2$ for some positive integer $a$.
  • The AM-GM and Trivial inequalities are more useful than you might imagine!
  • Memorize, memorize, memorize the following things:
  1. The trigonometric facts.
  2. Everything on the Combinatorics page.
  3. Integrals and derivatives, especially integrals.
  • Test your skills on practice AIMEs (<url>resources.php more resources</url>) often!

Back to Introduction



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