Difference between revisions of "1978 AHSME Problems/Problem 20"
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Latest revision as of 11:24, 24 March 2024
Problem 20
If 
are non-zero real numbers such that ![\[\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a},\]](http://latex.artofproblemsolving.com/9/7/3/9733280bec67e5196a2784002ad5f93db82b8d47.png)
and ![\[x=\frac{(a+b)(b+c)(c+a)}{abc},\]](http://latex.artofproblemsolving.com/1/a/8/1a811c35549cfac436bf409b437791e9f6cbab38.png)
and 
then 
equals
Solution
From the equation we add 
to each fraction to get ![\[\frac{a+b+c}{c}=\frac{a+b+c}{b}=\frac{a+b+c}{a}.\]](http://latex.artofproblemsolving.com/5/9/0/59023322e09a64cfb215ac0c0cdc98d676ea8cfc.png)
We perform casework on 
- If
then 
from which 
However, this contradicts the precondition 
then 
from which 
However, this contradicts the precondition 
- If
then 
then 
~MRENTHUSIASM
See also
| 1978 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
