Compute the value of
\[\frac{2}{1+\frac{1+0}{1+\frac{1+2}{1+5}}}\]Cities $A$ and $B$ are connected by a highway 100 miles long. Alice sets out from city $A$ towards $B$ at a constant rate of 20 miles per hour, while Bob sets out from city $B$ at $150%$ the speed of Alice. When the two meet, Bob speeds up to a constant $n$ miles per hour, while Alice continues towards $B$ at the same rate as before. (As soon as Bob reaches $A$, he turns around and sets back towards $B$.) Given that the two reach city $B$ at the same time, what is the numerical value of $n$? Round your answer to the nearest tenth.
A right triangle has legs of length $20$ and $21$. A circle of radius $2$ rolls about the outside perimeter of the triangle, always keeping tangent at one point. Suppose the area of the region “swept-out” by (only!) the rolling circle can be represented in the form $a+b\pi$, where both $a$ and $b$ are positive integers. Find $a+b$.
A rigged coin is such that flipping 4 heads in a row is 4 times as likely as flipping 4 tails in a row. The probability that one flips 2 tails and 2 heads in some order can be represented as $m-n\sqrt{2}$ for positive integers $m$ and $n$. Find $m$.
Recall that the $n$th triangular number is defined to be the sum of the first $n$ positive integers. Similarly, define the $n$th tetrahedral number as the sum of the first $n$ triangular numbers. How many $k$ exist between $1$ and $100$ (inclusive) such that the $k$th tetrahedral number is divisible by $9$?
Let $S$ be the list of (possibly repeated*) rational numbers formed such for each $10$-tuple of nonnegative integers $(a_1,a_2,a_3,\dots,a_{10})$, the number
\[n=\frac{1}{2^{a_1}3^{a_2}\cdots 11^{a_{10}}}\]is added to $S$. Find the sum of the elements in $S$.
*For example, the number $\frac{1}{4}$ is a result from both tuples $(2,0,0,0,0\dots,0)$ and $(0,0,1,0,0,\dots,0)$ and thus appears twice in the list.
Triangle $ABC$ is right at angle $B$. The midpoints of $AB$ and $BC$ are denoted as $M$ and $N$. Segments $AN$ and $CM$ intersect at point $P$ such that $PM=4$ and $PN=3$. Compute the square of the length of hypotenuse $\overline{AC}$.