Difference between revisions of "2012 AMC 12A Problems/Problem 20"
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<cmath>\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 24 </cmath> | <cmath>\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 24 </cmath> | ||
− | == Solution == | + | == Solution 1== |
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 <math>2</math> from each factor. | 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 <math>2</math> from each factor. | ||
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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 the answer is <math>6 \rightarrow \boxed{B}</math>. | Thus, the coefficient of the <math>x^{2012}</math> term is <math>32 \times 2 \times 1 = 64 = 2^6</math>. So the answer is <math>6 \rightarrow \boxed{B}</math>. | ||
+ | |||
+ | == Solution 2 == | ||
+ | The degree of <math>P(x)</math> is <math>1024+512+256+\cdots+1=2047</math>. We want to find the coefficient of <math>x^{2012}</math>, so we need to omit powers of <math>2</math> that add up to <math>2047-2012=35</math>. Because <math>35</math> is odd, we know that one of these must be <math>2^0</math>. Then, we can test all cases (there are very few of them) and we find that only <math>2^0+2^1+2^5</math> works. From here, we know that the answer is <math>2^0\cdot2^1\cdot2^5=2^6</math>. Therefore, the answer is <math>\boxed{(B)\:6.}</math> | ||
==See Also== | ==See Also== |
Revision as of 22:17, 9 February 2016
Contents
Problem
Consider the polynomial
The coefficient of is equal to . What is ?
Solution 1
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 from each factor.
Every number, including , 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 , , and .
Thus, from the factors, , , and were chosen as opposed to , and .
Thus, the coefficient of the term is . So the answer is .
Solution 2
The degree of is . We want to find the coefficient of , so we need to omit powers of that add up to . Because is odd, we know that one of these must be . Then, we can test all cases (there are very few of them) and we find that only works. From here, we know that the answer is . Therefore, the answer is
See Also
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.