1. Find the value of $\frac{(1 \times 2 \times 3)+(2 \times 4 \times 6)+(3 \times 6 \times 9)+\cdots+(335 \times 670 \times 1005)}{(1 \times 3 \times 6)+(2 \times 6 \times 12)+(3 \times 9 \times 18)+\cdots+(335 \times 1005 \times 2010)}$.
(A) $\frac{1}{3}$
(B) $\frac{2}{3}$
(C) $\frac{1}{6}$
(D) $\frac{1}{2}$
(E) $\frac{4}{9}$
1. Answer: (A)
$$
\begin{aligned}
& \frac{(1 \times 2 \times 3)+(2 \times 4 \times 6)+(3 \times 6 \times 9)+\cdots+(335 \times 670 \times 1005)}{(1 \times 3 \times 6)+(2 \times 6 \times 12)+(3 \times 9 \times 18)+\cdots+(335 \times 1005 \times 2010)} \\
= & \frac{(1 \times 2 \times 3)\left[1^{3}+2^{3}+3^{3}+\cdots+335^{3}\right]}{(1 \times 3 \times 6)\left[1^{3}+2^{3}+3^{3}+\cdots+335^{3}\right]} \\
= & \frac{1 \times 2 \times 3}{1 \times 3 \times 6}=\frac{1}{3} .
\end{aligned}
$$