2021 AMC 12A Fall Problem 12

Below is the professionally curated solution for Problem 12 of the 2021 AMC 12A Fall, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12A Fall solutions, or check the answer key.

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Concepts:binomial theorem divisibility

Difficulty rating: 1630

12.

What is the number of terms with rational coefficients among the $1001$ terms in the expansion of $\left(x\sqrt[3]{2} + y\sqrt{3}\right)^{1000}?$

$0$

$166$

$167$

$500$

$501$

Solution:

The general term is $\binom{1000}{k}(x\sqrt[3]{2})^{1000-k}(y\sqrt{3})^{k},$ whose coefficient contains $2^{(1000-k)/3}$ and $3^{k/2}.$ This is rational exactly when $3 \mid (1000 - k)$ and $k$ is even.

Since $1000 \equiv 1 \pmod 3,$ we need $k \equiv 1 \pmod 3$ and $k$ even, which combine to $k \equiv 4 \pmod 6.$ The valid values $k = 4, 10, \ldots, 1000$ number $\dfrac{1000 - 4}{6} + 1 = 167.$

Thus, the correct answer is C.

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