1975 AMC 12 Problem 5

Attempt Problem 5 of the 1975 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 1975 AMC 12 solutions, or check the answer key.

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

The polynomial

(x+y)^9

is expanded in decreasing powers of

x.

The second and third terms have equal values when evaluated at

x=p

and

y=q,

where

p

and

q

are positive numbers whose sum is one. What is the value of

p?

\frac15

\frac45

\frac14

\frac34

\frac89

Write the second and third binomial terms explicitly

After canceling common positive factors, combine

p=4q

with

p+q=1

The terms are

9p^8q

and

36p^7q^2.

Their equality, with

p,q\gt0,

gives

p=4q.

Since

p+q=1,

we get

5q=1

and

p=\frac{4}{5}.

Therefore, the correct answer is B.

Problem 5 in other years