1959 AMC 12 Problem 35

Question

The symbol

\ge

means “greater than or equal to”; the symbol

\le

means “less than or equal to.” In the equation

(x-m)^2-(x-n)^2=(m-n)^2,

m

is a fixed positive number, and

n

is a fixed negative number. The set of values

x

satisfying the equation is:

x\ge0

x\le n

x=0

the set of all real numbers

none of these

Factor the difference of squares on the left

Because

m\ne n,

divide by

m-n

and solve for

x

Factoring the left side gives

\begin{aligned} &(n-m)(2x-m-n)\\ &\qquad=(m-n)^2. \end{aligned}

Since

m\ne n,

division by

n-m

yields

2x-m-n=n-m,

so

x=n.

This single negative value is not any of choices A through D.

Therefore, the correct answer is E.

Problem 35 in other years