Lesson Progress
0% Complete

# Understand the problem

Suppose $a$, $b$, and $c$ are nonzero real numbers, and $a+b+c=0$. What are the possible value(s) for     $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}$      ? $\textbf{(A) }0\qquad\textbf{(B) }1\text{ and }-1\qquad\textbf{(C) }2\text{ and }-2\qquad\textbf{(D) }0,2,\text{ and }-2\qquad\textbf{(E) }0,1,\text{ and }-1$

# 2017 AMC 8 Problem 21

Number Theory
Easy
##### Suggested Book
Excursion in Mathematics

Do you really need a hint? Try it first!

Try to think elementarily . That means from the given relation $a+ b +c =0$ , what can be the possible signs of the real numbers , $a, b , \ and \ c$ accordingly. Then proceed .
There are $2$ cases to consider: Case $1$:  2 of $a$$b$, and $c$ are positive and the other is negative.  With out loss of generality, we can assume that $a$ and $b$ are positive and $c$ (as the relation is symmetric) is negative. In this case, we have that , $$\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}=1+1-1-1=0.$$ Think about the another similar case .
Case $2$ : of $a$, $b$, and $c$ are positive and the other is negative.  Here also without loss of generality, we can assume that $a$ and $b$ are negative and $c$ is positive. In this case, we have that $$\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}=-1-1+1+1=0.$$
In both cases, we get that the given expression equals    $\boxed{\textbf{(A)}\ 0}$.

# Connected Program at Cheenta

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

# Similar Problems

## Unit digit | Algebra | AMC 8, 2014 | Problem 22

Try this beautiful problem from Algebra about unit digit from AMC-8, 2014. You may use sequential hints to solve the problem.

## Problem based on Integer | PRMO-2018 | Problem 6

Try this beautiful problem from Algebra based on Quadratic equation from PRMO 8, 2018. You may use sequential hints to solve the problem.

## Number counting | ISI-B.stat Entrance | Objective from TOMATO

Try this beautiful problem Based on Number counting .You may use sequential hints to solve the problem.

## Area of a Triangle | AMC-8, 2000 | Problem 25

Try this beautiful problem from Geometry: Area of the triangle from AMC-8, 2000, Problem-25. You may use sequential hints to solve the problem.

## Mixture | Algebra | AMC 8, 2002 | Problem 24

Try this beautiful problem from Algebra based on mixture from AMC-8, 2002.. You may use sequential hints to solve the problem.

## Trapezium | Geometry | PRMO-2018 | Problem 5

Try this beautiful problem from Geometry based on Trapezium from PRMO , 2018. You may use sequential hints to solve the problem.

## Probability Problem | AMC 8, 2016 | Problem no. 21

Try this beautiful problem from Probability from AMC-8, 2016 Problem 21. You may use sequential hints to solve the problem.

## Pattern Problem| AMC 8, 2002| Problem 23

Try this beautiful problem from Pattern from AMC-8(2002) problem no 23.You may use sequential hints to solve the problem.

## Quadratic Equation Problem | PRMO-2018 | Problem 9

Try this beautiful problem from Algebra based on Quadratic equation from PRMO 8, 2018. You may use sequential hints to solve the problem.

## Set theory | ISI-B.stat Entrance | Objective from TOMATO

Try this beautiful problem Based on Set Theory .You may use sequential hints to solve the problem.