User:Temperal/The Problem Solver's Resource Tips and Tricks

The Problem Solver's Resource

Introduction \| Other Tips and Tricks \| Methods of Proof \| You are currently viewing the tips and ticks 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 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 $o$ , $o=2n\pm 1$ for some integer $n$ , and $o=a^2-(a-1)^2$ for some positive integer $a$ .