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

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==<span style="font-size:20px; color: blue;">Other Tips and Tricks</span>==
 
==<span style="font-size:20px; color: blue;">Other Tips and Tricks</span>==
 
This is a collection of general techniques for solving problems.
 
This is a collection of general techniques for solving problems.

Revision as of 20:11, 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 AMGM 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.

Back to Introduction



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