1994 AMC 12 Problem 25

Attempt Problem 25 of the 1994 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 1994 AMC 12 solutions, or check the answer key.

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

25.

x

and

y

are non-zero real numbers such that

|x|+y=3

and

|x|y+x^3=0,

then the integer nearest to

x-y

-3

-1

2

3

5

Consider the signs of

x

separately

For

x\lt0,

put

t=-x

and combine

y=t^2

with

t+y=3

x\gt0,

the second equation gives

y=-x^2,

while the first gives

y=3-x;

these would require

x^2-x+3=0,

which has no real root. Hence

x\lt0.

Put

t=-x\gt0.

Then the equations become

t+y=3

and

ty-t^3=0,

y=t^2.

Therefore

x-y=-t-t^2=-3,

already an integer. Thus the correct answer is A.

Problem 25 in other years