[u]Round 1[/u]


[b]p1.[/b] The temperature inside is $28^o$ F. After the temperature is increased by $5^o$ C, what will the new temperature in Fahrenheit be?


[b]p2.[/b] Find the least positive integer value of $n$ such that $\sqrt{2021+n}$ is a perfect square.


[b]p3.[/b] A heart consists of a square with two semicircles attached by their diameters as shown in the diagram. Given that one of the semicircles has a diameter of length $10$, then the area of the heart can be written as $a +b\pi$ where $a$ and $b$ are positive integers. Find $a +b$.
[img]https://cdn.artofproblemsolving.com/attachments/7/b/d277d9ebad76f288504f0d5273e19df568bc44.png[/img]

[u]Round 2[/u]


[b]p4.[/b] An $L$-shaped tromino is a group of $3$ blocks (where blocks are squares) arranged in a $L$ shape, as pictured below to the left. How many ways are there to fill a $12$ by $2$ rectangle of blocks (pictured below to the right) with $L$-shaped trominos if the trominos can be rotated or reflected?
[img]https://cdn.artofproblemsolving.com/attachments/d/c/cf37cdf9703ae0cd31c38af23b6874fddb3c12.png[/img]

[b]p5.[/b] How many permutations of the word $PIKACHU$ are there such that no two vowels are next to each other?


[b]p6.[/b] Find the number of primes $n$ such that there exists another prime $p$ such that both $n +p$ and $n-p$ are also prime numbers.


[u]Round 3[/u]


[b]p7.[/b] Maisy the Bear is at the origin of the Cartesian Plane. WhenMaisy is on the point $(m,n)$ then it can jump to either $(m,n +1)$ or $(m+1,n)$. Let $L(x, y)$ be the number of jumps it takes forMaisy to reach point (x, y). The sum of $L(x, y)$ over all lattice points $(x, y)$ with both coordinates between $0$ and $2020$, inclusive, is denoted as $S$. Find $\frac{S}{2020}$ .


[b]p8.[/b] A circle with center $O$ and radius $2$ and a circle with center $P$ and radius $3$ are externally tangent at $A$. Points $B$ and $C$ are on the circle with center $O$ such that $\vartriangle ABC$ is equilateral. Segment $AB$ extends past $B$ to point $D$ and $AC$ extends past $C$ to point $E$ such that $BD = CE = \sqrt3$. The area of $\vartriangle DEP$ can be written as $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers such that $b$ is squarefree and $gcd (a,c) = 1$. Find $a +b +c$.


[b]p9.[/b] Find the number of trailing zeroes at the end of $$\prod^{2021}_{i=1}(2021+i -1) = (2021)(2022)...(4041).$$


[u]Round 4[/u]


[b]p10.[/b] Let $a, b$, and $c$ be side lengths of a rectangular prism with space diagonal $10$. Find the value of $$(a +b)^2 +(b +c)^2 +(c +a)^2 -(a +b +c)^2.$$


[b]p11.[/b] In a regular heptagon $ABCDEFG$, $\ell$ is a line through $E$ perpendicular to $DE$. There is a point $P$ on $\ell$ outside the heptagon such that $PA = BC$. Find the measure of $\angle EPA$.


[b]p12.[/b] Dunan is being "$SUS$". The word "$SUS$" is a palindrome. Find the number of palindromes that can be written using some subset of the letters $\{S, U, S, S, Y, B, A, K, A\}$.


PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166494p28814284]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166500p28814367]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the problem, we need to determine the number of ways to fill a $12 \times 2$ rectangle using $L$-shaped trominos. Let's denote $a_n$ as the number of ways to tile a $2 \times n$ rectangle using $L$-shaped trominos.

1. **Define the recurrence relation:**
   We start by considering the possible placements of the first $L$-shaped tromino. Since the tromino covers 3 squares, the remaining part of the rectangle will be a $2 \times (n-3)$ rectangle. The tromino can be placed in two distinct orientations (rotated or reflected), so we have:
   \[
   a_n = 2a_{n-3} \quad \forall \quad n \geq 3
   \]

2. **Base cases:**
   We need to establish the initial conditions for the recurrence relation. For $n < 3$, it is impossible to place an $L$-shaped tromino, so:
   \[
   a_0 = 1, \quad a_1 = 0, \quad a_2 = 0
   \]

3. **Calculate the number of ways for $n = 12$:**
   Using the recurrence relation, we can compute $a_{12}$ step-by-step:
   \[
   a_{12} = 2a_9
   \]
   \[
   a_9 = 2a_6
   \]
   \[
   a_6 = 2a_3
   \]
   \[
   a_3 = 2a_0 = 2 \cdot 1 = 2
   \]
   Substituting back, we get:
   \[
   a_6 = 2a_3 = 2 \cdot 2 = 4
   \]
   \[
   a_9 = 2a_6 = 2 \cdot 4 = 8
   \]
   \[
   a_{12} = 2a_9 = 2 \cdot 8 = 16
   \]

Thus, the number of ways to fill a $12 \times 2$ rectangle using $L$-shaped trominos is $\boxed{16}$.