1978 AMC 12 Problem 6

Attempt Problem 6 of the 1978 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1978 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

The number of distinct pairs

(x,y)

of real numbers satisfying both of the following equations:

\begin{aligned} x&=x^2+y^2,\\ y&=2xy. \end{aligned}

0

1

2

3

4

Factor the second equation as

y(1-2x)=0

Handle

y=0

and

x=\frac12

as separate cases

y=0,

the first equation gives

x=x^2,

(x,y)=(0,0)

(1,0).

y\ne0,

the second equation gives

x=\frac{1}{2}.

The first then gives

y^2=\frac{1}{4},

producing

(\frac{1}{2},\frac{1}{2})

and

(\frac{1}{2},-\frac{1}{2}).

There are four pairs.

Therefore, the correct answer is E.

Problem 6 in other years