Solution to Olympiad level counting

In 3Blue1Brown’s latest video Olympiad level counting: How many subsets of {1,…,2000} have a sum divisible by 5?, you can find a few easy exercises at the end. These are the solutions.

Problem 1

\[\begin{align*} \frac{d}{dx} \left[ \frac{1}{1 - x} \right] &= \frac{d}{dx} \left[ \sum_{n=0}^{\infty} x^n \right] \\ \frac{1}{(1 - x)^2} &= \sum_{n=0}^{\infty} nx^{n-1} \end{align*}\]

Note that the first term of the sum is 0 so,

\[\sum_{n=0}^{\infty} nx^{n-1} = \sum_{n=1}^{\infty} nx^{n-1}.\]

There are infinitely many scenarios you can roll a die before being the number 1.

Number of rolls	Scenario	Probability
1	Roll 1	$1/6$
2	Roll 2-6, roll 1	$5/6 \cdot 1/6$
…
$n$	Roll 2-6 $n - 1$ times, roll 1	$(5/6)^{n-1} (1/6)$
…

Then the expected value of the number of rolls before seeing 1 is

\[\sum_{n=1}^{\infty} n p_n = \sum_{n=1}^{\infty} n (5/6)^{n-1} (1/6) = (1/6) \sum_{n=1}^{\infty} n(5/6)^{n-1}.\]

Let $x = 5/6$.

\[\sum_{n=1}^{\infty} nx^{n-1} = \frac{1}{(1 - x)^2} = \frac{1}{(1 - 5/6)^2} = \frac{1}{1/36} = 36\]

Thus the expected number of rolls before seeing 1 is $(1/6)36 = 6$.

Problem 2

Using the Binomial Theorem,

\[(1 + x)^n = \sum_{k=0}^{n} {n \choose k} x^k.\]

Letting $x = 2$,

\[\sum_{k=0}^{n} {n \choose k} 2^k = (1 + 2)^n = 3^n.\]

Problem 3

\[\begin{align*} F(x) &= \sum_{n=0}^{\infty} f_n \frac{x^n}{n!} \\ F'(x) &= \sum_{n=1}^{\infty} f_n \frac{x^{n-1}}{(n - 1)!} \\ F''(x) &= \sum_{n=2}^{\infty} f_n \frac{x^{n-2}}{(n - 2)!} \end{align*}\]

We can reindex the bottom 2 sums by placing $n$ by $n + 1$ and $n + 2$ respectively. \(\begin{align*} F'(x) &= \sum_{n=0}^{\infty} f_{n+1} \frac{x^n}{n!} \\ F''(x) &= \sum_{n=0}^{\infty} f_{n+2} \frac{x^n}{n!} \end{align*}\) Thus, \(F''(x) = \sum_{n=0}^{\infty} f_{n+2} \frac{x^n}{n!} = \sum_{n=0}^{\infty} (f_{n+1} + f_n) \frac{x^n}{n!} = F'(x) + F(x).\)

Bonus

\[\begin{align*} F''(x) &= F'(x) + F(x) \\ F''(x) - F'(x) - F(x) &= 0 \end{align*}\]

The characteristic of the homogeneous second order differential equation is $x^2 - x - 1 = 0$ which has roots

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}\]

Because there are 2 distinct real roots, the general solution is

\[F(x) = c_1e^{\frac{1 + \sqrt{5}}{2}x} + c_2 e^{\frac{1 - \sqrt{5}}{2}x}.\]

Since $F(0) = 0$ and $F’(0) = 1$,

\[\begin{align*} 0 &= c_1 + c_2 \\ 0 &= \frac{1 + \sqrt{5}}{2} c_1 + \frac{1 - \sqrt{5}}{2} c_2. \end{align*}\]

Thus,

\[\begin{align*} 1 &= \left( \frac{1 + \sqrt{5}}{2} - \frac{1 - \sqrt{5}}{2} \right) c_1 \\ &= \frac{1 + \sqrt{5} - 1 + \sqrt{5}}{2} c_1 \\ &= \frac{2\sqrt{5}}{2} c_1 \\ c_1 &= \frac{1}{\sqrt{5}} \\ c_2 &= -\frac{1}{\sqrt{5}} \end{align*}\]

Recall that $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$.

\[\begin{align*} F(x) &= \frac{1}{\sqrt{5}} \sum_{n=0}^{\infty} \frac{(\frac{1 + \sqrt{5}}{2})^n x^n}{n!} - \frac{1}{\sqrt{5}} \sum_{n=0}^{\infty} \frac{(\frac{1 - \sqrt{5}}{2})^n x^n}{n!} \\ &= \sum_{n=0}^{\infty} \frac{(\frac{1 + \sqrt{5}}{2})^n x^n - (\frac{1 - \sqrt{5}}{2})^n x^n}{\sqrt{5} n!} \\ &= \sum_{n=0}^{\infty} \frac{(\frac{1 + \sqrt{5}}{2})^n - (\frac{1 - \sqrt{5}}{2})^n}{\sqrt{5}} \frac{x^n}{n!} = \sum_{n=0}^{\infty} f_n \frac{x^n}{n!} \end{align*}\]

Hence,

\[f_n = \frac{1}{\sqrt{5}} \left[ \left( \frac{1 + \sqrt{5}}{2} \right)^n - \left( \frac{1 - \sqrt{5}}{2} \right)^n \right].\]

Problem 4

Recall from Problem 1 that

\[\frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n.\]

Thus,

\[\begin{align*} \frac{f(x)}{1 - x} &= \left( \sum_{n=0}^{\infty} a_n x^n \right) \left( \sum_{n=0}^{\infty} x^n \right) \\ &= \left( a_0 + a_1x + a_2x^2 + \cdots \right) \left( 1 + x + x^2 + \cdots \right) \\ &= a_0 + a_0x + a_0x^2 + \cdots \\ &\qquad \,\,\, + \, a_1x + a_1x^2 + \cdots \\ &\qquad\qquad\quad\, + \, a_2x^2 + \cdots \\ &= \sum_{n=0}^{\infty} S_n x^n \end{align*}\]

where $S_n = \sum_{k=0}^{n} a_k$ is the partial sum of the coefficients up to $a_n$.