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

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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 $\tan(x+y)$.


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?


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!

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