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

 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 $\tan x+\tan y=25$ 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 $a$.

### Example Problem 2

How many quadruples $(a,b,c,d)$ are there such that $a+b+c+d=98$ and $a,b,c,d$ are all odd?

#### Solution

Since they're odd, $a, b, c, d$ can each be expressed as $2n+1$ for some positive integer (or zero) $n$. Thus: $2n_1-1+2n_2-1+2n_3+1+2n_4+1=98$ $\Rightarrow 2(n_1+n_2+n_3+n_4)+4=98$ $\Rightarrow 2(n_1+n_2+n_3+n_4)=94$ $\Rightarrow n_1+n_2+n_3+n_4=47$ Binomial coefficients will yield the answer of $\boxed{19600}$.

• 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.

Remember, though, don't memorize without understanding!

• Test your skills on practice AIMEs (<url>resources.php more resources</url>) often!