[b]p1.[/b] Compute the greatest integer less than or equal to $$\frac{10 + 12 + 14 + 16 + 18 + 20}{21}$$


[b]p2.[/b] Let$ A = 1$.$B = 2$, $C = 3$, $...$, $Z = 26$. Find $A + B +M + C$.


[b]p3.[/b] In Mr. M's farm, there are $10$ cows, $8$ chickens, and $4$ spiders. How many legs are there (including Mr. M's legs)?


[b]p4.[/b] The area of an equilateral triangle with perimeter $18$ inches can be expressed in the form $a\sqrt{b}{c}$ , where $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Find $a + b + c$.


[b]p5.[/b] Let $f$ be a linear function so $f(x) = ax + b$ for some $a$ and $b$. If $f(1) = 2017$ and $f(2) = 2018$, what is $f(2019)$?


[b]p6.[/b] How many integers $m$ satisfy $4 < m^2 \le 216$?


[b]p7.[/b] Allen and Michael Phelps compete at the Olympics for swimming. Allen swims $\frac98$ the distance Phelps swims, but Allen swims in $\frac59$ of Phelps's time. If Phelps swims at a rate of $3$ kilometers per hour, what is Allen's rate of swimming? The answer can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$.


[b]p8.[/b] Let $X$ be the number of distinct arrangements of the letters in "POONAM," $Y$ be the number of distinct arrangements of the letters in "ALLEN" and $Z$ be the number of distinct arrangements of the letters in "NITHIN." Evaluate $\frac{X+Z}{Y}$ :


[b]p9.[/b] Two overlapping circles, both of radius $9$ cm, have centers that are $9$ cm apart. The combined area of the two circles can be expressed as $\frac{a\pi+b\sqrt{c}+d}{e}$ where $c$ is not divisible by the square of any prime and the fraction is simplified. Find $a + b + c + d + e$.


[b]p10.[/b] In the Boxborough-Acton Regional High School (BARHS), $99$ people take Korean, $55$ people take Maori, and $27$ people take Pig Latin. $4$ people take both Korean and Maori, $6$ people take both Korean and Pig Latin, and $5$ people take both Maori and Pig Latin. $1$ especially ambitious person takes all three languages, and and $100$ people do not take a language. If BARHS does not oer any other languages, how many students attend BARHS?


[b]p11.[/b] Let $H$ be a regular hexagon of side length $2$. Let $M$ be the circumcircle of $H$ and $N$ be the inscribed circle of $H$. Let $m, n$ be the area of $M$ and $N$ respectively. The quantity $m - n$ is in the form $\pi a$, where $a$ is an integer. Find $a$.


[b]p12.[/b] How many ordered quadruples of positive integers $(p, q, r, s)$ are there such that $p + q + r + s \le 12$?


[b]p13.[/b] Let $K = 2^{\left(1+ \frac{1}{3^2} \right)\left(1+ \frac{1}{3^4} \right)\left(1+ \frac{1}{3^8}\right)\left(1+ \frac{1}{3^{16}} \right)...}$. What is $K^8$?


[b]p14.[/b] Neetin, Neeton, Neethan, Neethine, and Neekhil are playing basketball. Neetin starts out with the ball. How many ways can they pass 5 times so that Neethan ends up with the ball?


[b]p15.[/b] In an octahedron with side lengths $3$, inscribe a sphere. Then inscribe a second sphere tangent to the first sphere and to $4$ faces of the octahedron. The radius of the second sphere can be expressed in the form $\frac{\sqrt{a}-\sqrt{b}}{c}$ , where the square of any prime factor of $c$ does not evenly divide into $b$. Compute $a + b + c$.


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the problem, we need to find the linear function \( f(x) = ax + b \) given the conditions \( f(1) = 2017 \) and \( f(2) = 2018 \).

1. **Set up the equations based on the given conditions:**
   \[
   f(1) = a \cdot 1 + b = 2017 \implies a + b = 2017
   \]
   \[
   f(2) = a \cdot 2 + b = 2018 \implies 2a + b = 2018
   \]

2. **Solve the system of linear equations:**
   Subtract the first equation from the second equation to eliminate \( b \):
   \[
   (2a + b) - (a + b) = 2018 - 2017
   \]
   \[
   2a + b - a - b = 1
   \]
   \[
   a = 1
   \]

3. **Substitute \( a = 1 \) back into the first equation to find \( b \):**
   \[
   a + b = 2017
   \]
   \[
   1 + b = 2017
   \]
   \[
   b = 2016
   \]

4. **Form the linear function \( f(x) \):**
   \[
   f(x) = ax + b = 1x + 2016 = x + 2016
   \]

5. **Calculate \( f(2019) \):**
   \[
   f(2019) = 2019 + 2016 = 4035
   \]

The final answer is \( \boxed{4035} \).