1979 AMC 12 Problem 26

Question

The function

f

satisfies the functional equation

f(x)+f(y)=f(x+y)-xy-1

for every pair

x,

y

of real numbers. If

f(1)=1,

then the number of integers

n\ne1

for which

f(n)=n

is

0

1

2

3

infinite

Set one variable equal to

1

to get a recurrence for consecutive integer inputs

Find

f(0)

, solve the recurrence for all integers, and then impose

f(n)=n

Setting

x=1

gives

f(y+1)-f(y)=y+2.

Setting

x=y=0

gives

f(0)=-1.

Extending the recurrence in both directions yields, for every integer

n,

f(n)=\frac{n^2+3n-2}{2}.

Thus

f(n)=n

is equivalent to

\begin{aligned} n^2+n-2&=(n+2)(n-1)\\ &=0. \end{aligned}

The integer solutions are

n=-2

and

n=1,

and after excluding

1

exactly one remains.

Therefore, the correct answer is B.

Problem 26 in other years