Difference between revisions of "2012 AMC 12A Problems/Problem 20"
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== Solution 2 == | == 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>. | + | 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 the powers of <math>2</math> that add up to <math>2047-2012=35</math>. We find that <math>35=2^0+2^1+2^5</math>. 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== |
Latest revision as of 19:59, 5 February 2019
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 the powers of that add up to . We find that . 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.