# Difference between revisions of "2012 AMC 12A Problems/Problem 20"

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− | + | == Problem == | |

+ | Consider the polynomial | ||

− | + | <cmath>P(x)=\prod_{k=0}^{10}=(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024)</cmath> | |

− | + | The coefficient of <math>x^{2012}</math> is equal to <math>2^a</math>. What is <math>a</math>? | |

− | + | <cmath>\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 24 </cmath> | |

− | Thus, the coefficient of the x^2012 term is 32 | + | == 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 <math>x</math> 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. <math>2012 = 11111011100_2</math>, meaning <math>2012 = 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4</math>. | ||

+ | |||

+ | Thus, the <math>x^{2012}</math> term was made by multiplying <math>x^1024</math> from the <math>(x^{1024} + 1024)</math> factor, <math>x^{512}</math> from the <math>(x^{512} + 512)</math> factor, and so on. The only numbers not used are 32, 2, and 1. | ||

+ | |||

+ | Thus, from the <math>(x^{32} + 32), (x^2+2), (x+1)</math> factors, 32, 2, and 1 were chosen as opposed to <math>x^{32}, x^2</math>, and <math>x</math>. | ||

+ | |||

+ | 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. |

## Revision as of 17:02, 16 February 2012

## Problem

Consider the polynomial

The coefficient of is equal to . What is ?

## 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.