2012 AMC 12A Problems/Problem 20: Difference between revisions
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Thus, the coefficient of the <math>x^{2012}</math> term is <math>32 \times 2 \times 1 = 64 = 2^6</math>. So, 6 is the right answer, or B. | Thus, the coefficient of the <math>x^{2012}</math> term is <math>32 \times 2 \times 1 = 64 = 2^6</math>. So, 6 is the right answer, or B. | ||
{{AMC12 box|year=2012|ab=A|num-b=19|num-a=21}} | |||
Revision as of 14:49, 19 February 2012
Problem
Consider the polynomial
![\[P(x)=\prod_{k=0}^{10}=(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024)\]](http://latex.artofproblemsolving.com/9/e/c/9ec81d75d7273f6b19ccee6b2771dc7f41a787a6.png)
The coefficient of 
is equal to 
. What is 
?
![\[\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 24\]](http://latex.artofproblemsolving.com/f/f/0/ff00a33605701fb3b8adff21802c46d01211a2ed.png)
Solution
Every term in the expansion of the product is formed by taking one term from each factor and multiplying them all together. Therefore, we pick a power of 
or a power of 2 from each factor.
Every number, including 2012, has a unique representation by the sum of powers of two, and that representation can be found by converting a number to its binary form. 
, meaning 
.
Thus, the term was made by multiplying 
from the 
factor, 
from the 
factor, and so on. The only numbers not used are 32, 2, and 1.
Thus, from the 
factors, 32, 2, and 1 were chosen as opposed to 
, and .
Thus, the coefficient of the term is 
. So, 6 is the right answer, or B.
| 2012 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 19 |
Followed by Problem 21 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
