# Difference between revisions of "2017 AMC 8 Problems/Problem 21"

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+ | ==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>? | ||

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+ | <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> | ||

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+ | ==Solution== | ||

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There are <math>2</math> cases to consider: | There are <math>2</math> cases to consider: | ||

## Revision as of 13:34, 22 November 2017

## Problem 21

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. WLOG 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 assume that and are negative and is positive. In this case, we have that

In both cases, we get that the given expression equals .

~nukelauncher