Determine the fraction of the amount remaining each month.

Each month, half of the remaining amount is given away, so the fraction remaining is $\frac{1}{2}$.

After 6 months, the fraction remaining will be $\left(\frac{1}{2}\right)^6$.

The fraction remaining after 6 months is $\boxed{\frac{1}{64}}$.

After 12 months, the fraction remaining will be $\left(\frac{1}{2}\right)^{12}$.

The fraction remaining after 12 months is $\boxed{\frac{1}{4096}}$.

The fractions as powers of 2 with negative exponents are $\boxed{2^{-6}}$ and $\boxed{2^{-12}}$.

The initial amount is \1,000,000$. The end of the year means after 12 months. From part b), the fraction remaining is$\frac{1}{4096}$.

$\text{Remaining amount} = \$1\,000\,000 \times \frac{1}{4096}$ $= \frac{\$1\,000\,000}{4096}$ $= \$244.140625$

$\approx \$244.14$ The amount remaining at the end of the year is \boxed{\244.14}$.