1977 AHSME Problems/Problem 8

Problem 8

For every triple $(a,b,c)$ of non-zero real numbers, form the number $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}$ . The set of all numbers formed is

$\textbf{(A)}\ {0} \qquad \textbf{(B)}\ \{-4,0,4\} \qquad \textbf{(C)}\ \{-4,-2,0,2,4\} \qquad \textbf{(D)}\ \{-4,-2,2,4\}\qquad \textbf{(E)}\ \text{none of these}$

Solution

Solution by e_power_pi_times_i

$\dfrac{x}{|x|} = 1$ or $-1$ depending whether $x$ is positive or negative. If $a$ , $b$ , and $c$ are positive, then the entire thing amounts to $4$ . If one of the three is negative and the other two positive, the answer is $0$ . If two of the three is negative and one is positive, the answer is $0$ . If all three are negative, the answer is $-4$ . Therefore the set is $\boxed{\textbf{(B)}\ \{-4,0,4\}}$ .

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1977 AHSME Problems/Problem 8

Problem 8

Solution