Difference between revisions of "User:Temperal/The Problem Solver's Resource Tips and Tricks"
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| − | + | {{User:Temperal/testtemplate|the tips and tricks section}} | |
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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. | ||
*Don't be afraid to use casework! Sometimes it's the only way. (But be VERY afraid to use brute force.) | *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: | *Remember that substitution is a useful technique! Example problem: | ||
| − | ===Example Problem | + | ===Example Problem 1=== |
| − | If <math>\tan x+\tan y=25</math> and <math>\cot x+\ cot y=30</math>, find <math>\tan(x+y)</math>. | + | If <math>\tan x+\tan y=25</math> and <math>\cot x+\cot y=30</math>, find <math>\tan(x+y)</math>. |
====Solution==== | ====Solution==== | ||
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>. | ||
| − | 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. | + | 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. |
---- | ---- | ||
| − | *Remember the special properties of odd numbers: For any odd number <math>o</math>, <math>o=2n | + | *Remember the special properties of odd numbers: For any odd number <math>o</math>, <math>o=2n\pm 1</math> for some integer <math>n</math>, and <math>o=a^2-(a-1)^2</math> for some positive integer <math>a</math>. |
| − | *The | + | ===Example Problem 2=== |
| + | How many quadruples <math>(a,b,c,d)</math> are there such that <math>a+b+c+d=98</math> and <math>a,b,c,d</math> are all odd? | ||
| + | |||
| + | ====Solution==== | ||
| + | Since they're odd, <math>a, b, c, d</math> can each be expressed as <math>2n+1</math> for some positive integer (or zero) <math>n</math>. | ||
| + | Thus: | ||
| + | <math>2n_1-1+2n_2-1+2n_3+1+2n_4+1=98</math> | ||
| + | |||
| + | <math>\Rightarrow 2(n_1+n_2+n_3+n_4)+4=98</math> | ||
| + | |||
| + | <math>\Rightarrow 2(n_1+n_2+n_3+n_4)=94</math> | ||
| + | |||
| + | <math>\Rightarrow n_1+n_2+n_3+n_4=47</math> | ||
| + | Binomial coefficients will yield the answer of <math>\boxed{19600}</math>. | ||
| + | |||
| + | ---- | ||
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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. | ||
| − | *Test your skills on [ | + | Remember, though, don't memorize without understanding! |
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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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Latest revision as of 19:06, 10 January 2009
| 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 
and 
, find 
.
Solution
Let 
, 
. Thus, 
, 
, so 
, hence 
, which turns out to be 
.
This technique can also be used to solve quadratics of high degrees, i.e. 
; let 
, and solve from there.
- Remember the special properties of odd numbers: For any odd number
, 
for some integer 
, and 
for some positive integer 
.
, 
for some integer 
for some positive integer 
.Example Problem 2
How many quadruples 
are there such that 
and 
are all odd?
Solution
Since they're odd, 
can each be expressed as 
for some positive integer (or zero) 
.
Thus:
Binomial coefficients will yield the answer of 
.
- The AM-GM and Trivial inequalities are more useful than you might imagine!
- Memorize, memorize, memorize the following things:
- The trigonometric facts.
- Everything on the Combinatorics page.
- 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!
