Difference between revisions of "1978 AHSME Problems/Problem 20"
MRENTHUSIASM (talk | contribs) (The solution has some flaws. If b=c, then it is possible for (a,b,c)=(-2t,t,t), from which we obtain the correct answer -1. I will rewrite a solution with complete symmetry.) |
MRENTHUSIASM (talk | contribs) |
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==Solution== | ==Solution== | ||
| − | <b> | + | From the equation <cmath>\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a},</cmath> we add <math>2</math> to each fraction to get <cmath>\frac{a+b+c}{c}=\frac{a+b+c}{b}=\frac{a+b+c}{a}.</cmath> |
| + | We perform casework on <math>a+b+c:</math> | ||
| + | |||
| + | * If <math>a+b+c\neq0,</math> then <math>a=b=c,</math> from which <math>x=\frac{(2a)(2a)(2a)}{a^3}=8.</math> | ||
| + | |||
| + | * If <math>a+b+c=0,</math> then | ||
~MRENTHUSIASM | ~MRENTHUSIASM | ||
Revision as of 22:10, 4 September 2021
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 
then 
from which 
- 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. 
