Let $a_1,a_2,a_3,\dots$ be a sequence of numbers such that $a_{n+2} = 2a_n$ for all integers $n.$ Suppose $a_1 = 1,$ $a_2 = 3,$ \[ \sum_{n=1}^{2021} a_{2n} = c, \quad \text{and} \quad \sum\limits_{n=1}^{2021} a_{2n-1} = b. \] Then $c - b + \tfrac{c-b}{b}$ can be written in the form $x^y,$ where $x$ and $y$ are integers such that $x$ is as small as possible. Find $x + y.$

Big Chungus has been thinking of a new symbol for BMT, and the drawing below is what he came up with. If each of the $16$ small squares in the grid are unit squares, what is the area of the shaded region?

At a prom, there are $4$ boys and $3$ girls. Each boy picks a girl to dance with, and each girl picks a boy to dance with. Assuming that each choice is uniformly random, the probability that at least one boy and one girl choose each other as dance partners is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Compute $p+q.$

Carter and Vivian decide to spend their afternoon listing pairs of real numbers, $(a, b).$ Carter wants to find all $(a, b)$ such that $(a, b)$ lie within a circle of radius $6$ centered at $(6, 6).$ Vivian hates circles and would rather find all $(a, b)$ such that $a,$ $b,$ and $6$ can be the side lengths of a triangle. If Carter randomly chooses an $(a, b)$ that satisfies his conditions, then the probability that the pair also satisfies Vivian's conditions can be written in the form $\tfrac{p}{q} + \tfrac{r}{s\pi},$ where $p,$ $q,$ $r,$ and $s$ are positive integers, $p$ and $q$ are relatively prime, and $r$ and $s$ are relatively prime. Find $p + q + r + s.$

Keith decides that a sequence of digits is [i]slick[/i] if every pair of adjacent digits in the sequence is divisible by either $23$ or $17.$ What is the greatest possible number of $2$s in a $2021$-digit long slick sequence?

A pharmaceutical company produces a disease test that has a $95\%$ accuracy rate on individuals who actually have an infection, and a $90\%$ accuracy rate on individuals who do not have an infection. They use their test on a population of mathletes, of which $2\%$ actually have an infection. If a test concludes that a mathlete has an infection, then the probability that the mathlete actually does have an infection is $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Find $a + b.$

Let $ABCD$ be a rectangle with diagonals of length $10.$ Let $P$ be the midpoint of $\overline{AD},$ $S$ be the midpoint of $\overline{BC},$ and $T$ be the midpoint of $\overline{CD}.$ Points $Q$ and $R$ are chosen on $\overline{AB}$ such that $AP=AQ$ and $BR=BS,$ and minor arcs $\widehat{PQ}$ and $\widehat{RS}$ centered at $A$ and $B,$ respectively, are drawn. Circle $\omega$ is tangent to $\overline{CD}$ at $T$ and externally tangent to $\widehat{PQ}$ and $\widehat{RS}.$ Suppose that the radius of $\omega$ is $\tfrac{43}{18}.$ Then the sum of all possible values of the area of $ABCD$ can be written in the form $\tfrac{a+b\sqrt{c}}{d},$ where $a,\ b,\ c,$ and $d$ are positive integers, $b$ and $d$ are relatively prime, and $c$ is prime. Find $a+b+c+d.$

Squares of side lengths $1,$ $2,$ $3,$ and $4,$ are placed on a line segment $\ell$ from left to right, respectively, and these squares lie on the same side of $\ell,$ forming a polygon $P.$ An equilateral triangle whose base is $\ell$ is drawn around the squares such that its other two sides intersect $P$ at its leftmost and rightmost vertices (that are not on $\ell$). The area of the triangle can be written in the form $\tfrac{a + b\sqrt{3}}{c},$ where $a,$ $b,$ and $c$ are positive integers, and $b$ and $c$ are relatively prime. Find $a + b + c.$

Let $P$ and $Q$ be the midpoints of sides $AB$ and $BC,$ respectively, of $\triangle ABC.$ Suppose $\angle A = 30^{\circ}$ and $\angle PQC = 110^{\circ}.$ Find $\angle B$ in degrees.

Sasha has a bag that holds $6$ red marbles and $7$ green marbles. How many ways can Sasha pick a handful of (zero or more) marbles from the bag such that her handful contains at least as many red marbles as green marbles (any two marbles are distinguishable, even if they have the same color)?

Walter owns $11$ dumbbells, which have weights, in pounds, $1,$ $2,$ $\ldots,$ $10,$ $11.$ Walter wants to split his dumbbells into three groups of equal total weight. What is the smallest possible product that the dumbbell weights in any one of these groups can have?

Terry decides to practice his arithmetic by adding the numbers between $10$ and $99$ inclusive. However, he accidentally swaps the digits of one of the numbers, and thus gets the incorrect sum of $4941.$ What is the largest possible number whose digits Terry could have swapped in the summation?

Alice and Bob are put in charge of building a bridge with their respective teams. With both teams' combined effort, the team can be finished in $6$ days. In reality, Alice's team works alone for the first $3$ days, and then, they decide to take a break. Bob's team takes over from there and works for another $4$ days. As a result, $60\%$ of the bridge is successfully constructed. How many days would it take for Alice's team alone to finish building the bridge completely from the start?