Question

Consider the polynomial $P(x) = \prod_{k=0}^{10}\left(x^{2^k} + 2^k\right) = (x+1)(x^2+2)(x^4+4)\cdots(x^{1024}+1024).$

The coefficient of $x^{2012}$ is equal to $2^a.$ What is $a?$

$5$

$6$

$7$

$10$

$24$

Solution:

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.

2012 AMC 12A Problem 20

Solution:

Problem 20 in Other Years