1989 AMC 12 Problem 8

Attempt Problem 8 of the 1989 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 1989 AMC 12 solutions, or check the answer key.

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

For how many integers

n

between

1

and

100

does

x^2+x-n

factor into the product of two linear factors with integer coefficients?

0

1

2

9

10

Write the factors as

(x-r)(x+s)

with positive integers

r,s

The coefficient of

x

forces the two integer parameters to differ by

1

An integer factorization must have the form

(x-k)(x+k+1)

for some positive integer

k.

Expanding gives

x^2+x-k(k+1),

n=k(k+1).

The values

k=1,2,\ldots,9

give

n\le100,

while

10\cdot11\gt100.

There are

9

values.

Thus the correct answer is D.

Problem 8 in other years