[u]Round 1[/u]


[b]p1.[/b] Let $ABCDEF$ be a regular hexagon. How many acute triangles have all their vertices among the vertices of $ABCDEF$?


[b]p2.[/b] A rectangle has a diagonal of length $20$. If the width of the rectangle is doubled, the length of the diagonal becomes $22$. Given that the width of the original rectangle is $w$, compute $w^2$.


[b]p3.[/b] The number $\overline{2022A20B22}$ is divisible by 99. What is $A + B$?


[u]Round 2[/u]


[b]p4.[/b] How many two-digit positive integers have digits that sum to at least $16$?


[b]p5.[/b] For how many integers $k$ less than $10$ do there exist positive integers x and y such that $k =x^2 - xy + y^2$?


[b]p6.[/b] Isosceles trapezoid $ABCD$ is inscribed in a circle of radius $2$ with $AB \parallel CD$, $AB = 2$, and one of the interior angles of the trapezoid equal to $110^o$. What is the degree measure of minor arc $CD$?


[u]Round 3[/u]


[b]p7.[/b] In rectangle $ALEX$, point $U$ lies on side $EX$ so that $\angle AUL = 90^o$. Suppose that $UE = 2$ and $UX = 12$. Compute the square of the area of $ALEX$.


[b]p8.[/b] How many digits does $20^{22}$ have?


[b]p9.[/b] Compute the units digit of $3 + 3^3 + 3^{3^3} + ... + 3^{3^{...{^3}}}$ , where the last term of the series has $2022$ $3$s.


[u]Round 4[/u]


[b]p10.[/b] Given that $\sqrt{x - 1} + \sqrt{x} = \sqrt{x + 1}$ for some real number $x$, the number $x^2$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.


[b]p11.[/b]  Eric the Chicken Farmer arranges his $9$ chickens in a $3$-by-$3$ grid, with each chicken being exactly one meter away from its closest neighbors. At the sound of a whistle, each chicken simultaneously chooses one of its closest neighbors at random and moves $\frac12$ of a unit towards it. Given that the expected number of pairs of chickens that meet can be written as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers, compute $p + q$.


[b]p12.[/b] For a positive integer $n$, let $s(n)$ denote the sum of the digits of $n$ in base $10$. Find the greatest positive integer $n$ less than $2022$ such that $s(n) = s(n^2)$.


PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2949432p26408285]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. Let the width of the original rectangle be \( w \) and the length be \( l \). According to the Pythagorean theorem, the diagonal of the rectangle can be expressed as:
   \[
   w^2 + l^2 = 20^2
   \]
   Simplifying, we get:
   \[
   w^2 + l^2 = 400
   \]

2. When the width of the rectangle is doubled, the new width becomes \( 2w \). The new diagonal is given as 22. Using the Pythagorean theorem again, we have:
   \[
   (2w)^2 + l^2 = 22^2
   \]
   Simplifying, we get:
   \[
   4w^2 + l^2 = 484
   \]

3. We now have a system of equations:
   \[
   \begin{cases}
   w^2 + l^2 = 400 \\
   4w^2 + l^2 = 484
   \end{cases}
   \]

4. Subtract the first equation from the second equation to eliminate \( l^2 \):
   \[
   (4w^2 + l^2) - (w^2 + l^2) = 484 - 400
   \]
   Simplifying, we get:
   \[
   3w^2 = 84
   \]

5. Solving for \( w^2 \):
   \[
   w^2 = \frac{84}{3} = 28
   \]

The final answer is \( \boxed{28} \).