Difference between revisions of "2021 Fall AMC 12A Problems/Problem 12"
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<math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 166 \qquad\textbf{(C)}\ 167 \qquad\textbf{(D)}\ 500 \qquad\textbf{(E)}\ 501</math> | <math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 166 \qquad\textbf{(C)}\ 167 \qquad\textbf{(D)}\ 500 \qquad\textbf{(E)}\ 501</math> | ||
− | ==Solution | + | ==Solution== |
By the Binomial Theorem, each term in the expansion is of the form <cmath>\binom{1000}{k}\left(x\sqrt[3]{2}\right)^k\left(y\sqrt{3}\right)^{1000-k}=\binom{1000}{k}2^{\frac k3}3^{\frac{1000-k}{2}}x^k y^{1000-k},</cmath> where <math>k\in\{0,1,2,\ldots,1000\}.</math> | By the Binomial Theorem, each term in the expansion is of the form <cmath>\binom{1000}{k}\left(x\sqrt[3]{2}\right)^k\left(y\sqrt{3}\right)^{1000-k}=\binom{1000}{k}2^{\frac k3}3^{\frac{1000-k}{2}}x^k y^{1000-k},</cmath> where <math>k\in\{0,1,2,\ldots,1000\}.</math> | ||
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~MRENTHUSIASM | ~MRENTHUSIASM | ||
+ | ==Video Solution by TheBeautyofMath== | ||
+ | https://youtu.be/ToiOlqWz3LY?t=169 | ||
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==See Also== | ==See Also== | ||
{{AMC12 box|year=2021 Fall|ab=A|num-b=11|num-a=13}} | {{AMC12 box|year=2021 Fall|ab=A|num-b=11|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 21:01, 7 April 2022
Problem
What is the number of terms with rational coefficients among the terms in the expansion of
Solution
By the Binomial Theorem, each term in the expansion is of the form where
This problem is equivalent to counting the values of such that both and are integers. Note that must be a multiple of and a multiple of so must be a multiple of There are such values of
~MRENTHUSIASM
Video Solution by TheBeautyofMath
https://youtu.be/ToiOlqWz3LY?t=169
~IceMatrix
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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.