1959 AMC 12 Problem 48

Question

Given the polynomial

\begin{aligned} &a_0x^n+a_1x^{n-1}+\cdots\\ &\qquad{}+a_{n-1}x+a_n, \end{aligned}

where

n

is a positive integer or zero, and

a_0

is a positive integer. The remaining

a

’s are integers or zero. Set

\begin{aligned} h={}&n+a_0+|a_1|+|a_2|\\ &{}+\cdots+|a_n|. \end{aligned}

[See example

25

for the meaning of

|x|.

] The number of polynomials with

h=3

is:

3

5

6

7

9

Since

n+a_0\le3

and

a_0\ge1,

consider only

n=0,1,2

For each degree, count the integer coefficient tuples with the required sum of absolute values

We count by degree. If

n=0,

then

a_0=3,

giving one polynomial. If

n=1,

then

a_0+|a_1|=2.

This gives

a_0=2,a_1=0

or

a_0=1,a_1=\pm1,

for three polynomials. If

n=2,

then

a_0=1

and

a_1=a_2=0,

giving one more. No higher degree is possible. The total is

1+3+1=5.

Thus, the correct answer is B.

Problem 48 in other years