2017 AMC 8

Complete problem set with solutions and individual problem pages

Problem 21 Hard

Suppose a, b, and c are nonzero real numbers, and a+b+c=0. What are the possible value(s) for \frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}?

  • A.

    0

  • B.

    1\text{ and }-1

  • C.

    2\text{ and }-2

  • D.

    0,2,\text{ and }-2

  • E.

    0,1,\text{ and }-1

Answer:A

There are 2 cases to consider:

Case 1: 2 of a, b, and c are positive and the other is negative. Without loss of generality (WLOG), we can assume that a and b are positive and c is negative. In this case, we have that

\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}=1+1-1-1=0.

Case 2: 2 of a, b, and c are negative and the other is positive. WLOG, we can assume that a and b are negative and c is positive. In this case, we have that

\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}=-1-1+1+1=0.

Note these are the only valid cases, for neither 3 negatives nor 3 positives would work as they cannot sum up to 0. In both cases, we get that the given expression equals \boxed{\textbf{(A)}\ 0}.

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