Difference between revisions of "2017 AMC 8 Problems/Problem 21"
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Suppose <math>a</math>, <math>b</math>, and <math>c</math> are nonzero real numbers, and <math>a+b+c=0</math>. What are the possible value(s) for <math>\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}</math>? | Suppose <math>a</math>, <math>b</math>, and <math>c</math> are nonzero real numbers, and <math>a+b+c=0</math>. What are the possible value(s) for <math>\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}</math>? | ||
− | <math>\ | + | <math>\textbf{(A) }0\qquad\textbf{(B) }1\text{ and }-1\qquad\textbf{(C) }2\text{ and }-2\qquad\textbf{(D) }0,2,\text{ and }-2\qquad\textbf{(E) }0,1,\text{ and }-1</math> |
==Solution== | ==Solution== |
Latest revision as of 15:11, 2 March 2024
Contents
[hide]Problem
Suppose , , and are nonzero real numbers, and . What are the possible value(s) for ?
Solution
There are cases to consider:
Case : of , , and are positive and the other is negative. Without loss of generality (WLOG), we can assume that and are positive and is negative. In this case, we have that
Case : of , , and are negative and the other is positive. WLOG, we can assume that and are negative and is positive. In this case, we have that
Note these are the only valid cases, for neither negatives nor positives would work as they cannot sum up to . In both cases, we get that the given expression equals .
Video Solution (CREATIVE THINKING + ANALYSIS!!!)
~Education, the Study of Everything
Video Solution by OmegaLearn
https://youtu.be/7an5wU9Q5hk?t=2362
Video Solutions
- Happytwin
~savannahsolver
https://www.youtube.com/watch?v=u6M6ECGok1o
- SUS
See Also
2017 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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 | ||
All AJHSME/AMC 8 Problems and Solutions |