2012 AMC 12A Problem 20

Question

Consider the polynomial

\begin{aligned} P(x) &= \prod_{k=0}^{10}\left(x^{2^k} + 2^k\right) \\ &= (x+1)(x^2+2)(x^4+4) \\ &\quad \cdots (x^{1024}+1024). \end{aligned}

The coefficient of

x^{2012}

is equal to

2^a.

What is

a?

5

6

7

10

24

Each factor contributes either its

x^{2^k}

term or its constant

2^k

; the chosen exponents must sum to

2012

There is only one way to write

2012

as a sum of distinct powers of two, so exactly one term produces

x^{2012}

Expanding the product, a term of degree

2012

comes from choosing

x^{2^k}

from some factors so that the exponents sum to

2012.

Since powers of two are distinct, this corresponds to the binary representation

2012 = 11111011100_2.

That representation is unique, so exactly one term gives

x^{2012},

and its coefficient is the product of the constants

2^k

from the remaining factors: those with

k \in \{0, 1, 5\}.

The coefficient is

2^0 \cdot 2^1 \cdot 2^5 = 2^6,

so

a = 6.

Thus, the correct answer is B.

Problem 20 in other years