HMMT November, 2018

Solution:

Let $d_{x}$ be the number ways to make exactly $x$ cents using only dimes and nickels. It is easy to see that when $x$ is a multiple of $5$,
$$d_{x} = \left\lfloor \frac{x}{10} \right\rfloor + 1$$
Now, let $c_{x}$ be the number of ways to make exactly $x$ cents using only quarters, dimes and nickels. Again, it is easy to see that when $x$ is a multiple of $5$,
$$c_{x} = c_{x-25} + d_{x}$$
(We can either use 1 or more quarters, which corresponds to the $c_{x-25}$ term, or we can use 0 quarters, which corresponds to the $d_{x}$ term.) Combining these two equations, we see that $c_{x}$ can be approximated by a polynomial of degree 2. (In fact, we get five different approximations of $c_{x}$, depending on the value of $x \bmod 25$, but they all only differ by a constant, which will not affect the limit case.) We also see that
$$B_{n} = c_{100 n} + c_{100(n-1)} + \ldots + c_{0}$$
and
$$P_{n} = c_{100 n} + c_{100 n - 5} + \ldots + c_{0}$$
Suppose $a$ is the value such that $\lim_{n \rightarrow \infty} \frac{c_{n}}{a n^{2}} = 1$. Then
$$\lim_{n \rightarrow \infty} \frac{B_{n}}{P_{n}} = \lim_{n \rightarrow \infty} \frac{\sum_{k=0}^{\lfloor n / 100 \rfloor} a (100k)^{2}}{\sum_{k=0}^{\lfloor n / 5 \rfloor} a (5k)^{2}} = \lim_{n \rightarrow \infty} \frac{400 \cdot \frac{n}{100} \left( \frac{n}{100} + 1 \right) \left( 2 \cdot \frac{n}{100} + 1 \right)}{\frac{n}{5} \left( \frac{n}{5} + 1 \right) \left( 2 \cdot \frac{n}{5} + 1 \right)} = 20.$$