[b]p1.[/b] Calculate $A \cdot B +M \cdot C$, where $A = 1$, $B = 2$, $C = 3$, $M = 13$.


[b]p2.[/b] What is the remainder of $\frac{2022\cdot2023}{10}$ ?


[b]p3.[/b] Daniel and Bryan are rolling fair $7$-sided dice. If the probability that the sum of the numbers that Daniel and Bryan roll is greater than $11$ can be represented as the fraction $\frac{a}{b}$ where $a$, $b$ are relatively prime positive integers, what is $a + b$?


[b]p4.[/b] Billy can swim the breaststroke at $25$ meters per minute, the butterfly at $30$ meters per minute, and the front crawl at $40$ meters per minute. One day, he swam without stopping or slowing down, swimming $1130$ meters. If he swam the butterfly for twice as long as the breaststroke, plus one additional minute, and the front crawl for three times as long as the butterfly, minus eight minutes, for how many minutes did he swim?


[b]p5.[/b] Elon Musk is walking around the circumference of Mars trying to find aliens. If the radius of Mars is $3396.2$ km and Elon Musk is $73$ inches tall, the difference in distance traveled between the top of his head and the bottom of his feet in inches can be expressed as $a\pi$ for an integer $a$. Find $a$. ($1$ yard is exactly $0.9144$ meters).


[b]p6.[/b] Lukas is picking balls out of his five baskets labeled $1$,$2$,$3$,$4$,$5$. Each basket has $27$ balls, each labeled with the number of its respective basket. What is the least number of times Lukas must take one ball out of a random basket to guarantee that he has chosen at least $5$ balls labeled ”$1$”? If there are no balls in a chosen basket, Lukas will choose another random basket.


[b]p7.[/b] Given $35_a = 42_b$, where positive integers $a$, $b$ are bases, find the minimum possible value of the sum $a + b$ in base $10$.


[b]p8.[/b] Jason is playing golf. If he misses a shot, he has a $50$ percent chance of slamming his club into the ground. If a club is slammed into the ground, there is an $80$ percent chance that it breaks. Jason has a $40$ percent chance of hitting each shot. Given Jason must successfully hit five shots to win a prize, what is the expected number of clubs Jason will break before he wins a prize?


[b]p9.[/b] Circle $O$ with radius $1$ is rolling around the inside of a rectangle with side lengths $5$ and $6$. Given the total area swept out by the circle can be represented as $a + b\pi$ for positive integers $a$, $b$ find $a + b$. 


[b]p10.[/b] Quadrilateral $ABCD$ has $\angle ABC = 90^o$, $\angle ADC = 120^o$, $AB = 5$, $BC = 18$, and $CD = 3$. Find $AD$.


[b]p11.[/b] Raymond is eating huge burgers. He has been trained in the art of burger consumption, so he can eat one every minute. There are $100$ burgers to start with. However, at the end of every $20$ minutes, one of Raymond’s friends comes over and starts making burgers. Raymond starts with $1$ friend. If each of his friends makes $1$ burger every $20$ minutes, after how long in minutes will there be $0$ burgers left for the first time?


[b]p12.[/b] Find the number of pairs of positive integers $(a, b)$ and $b\le  a \le  2022$ such that $a\cdot lcm(a, b) = b \cdot gcd(a, b)^2$.


[b]p13.[/b] Triangle $ABC$ has sides $AB = 6$, $BC = 10$, and $CA = 14$. If a point $D$ is placed on the opposite side of $AC$ from $B$ such that $\vartriangle ADC$ is equilateral, find the length of $BD$.


[b]p14.[/b] If the product of all real solutions to the equation $(x-1)(x-2)(x-4)(x-5)(x-7)(x-8) = -x^2+9x-64$ can be written as $\frac{a-b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, b, d) = 1$ and $c$ is squarefree, compute $a + b + c + d$.


[b]p15.[/b] Joe has a calculator with the keys $1, 2, 3, 4, 5, 6, 7, 8, 9,+,-$. However, Joe is blind. If he presses $4$ keys at random, and the expected value of the result can be written as $\frac{x}{11^4}$ , compute the last $3$ digits of $x$ when $x$ divided by $1000$. (If there are consecutive signs, they are interpreted as the sign obtained when multiplying the two signs values together, e.g $3$,$+$,$-$,$-$, $2$ would return $3 + (-(-(2))) = 3 + 2 = 5$. Also, if a sign is pressed last, it is ignored.)


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Given the problem \(35_a = 42_b\), where \(a\) and \(b\) are bases, we need to find the minimum possible value of the sum \(a + b\) in base 10.

1. **Convert \(35_a\) to base 10:**
   \[
   35_a = 3a + 5
   \]

2. **Convert \(42_b\) to base 10:**
   \[
   42_b = 4b + 2
   \]

3. **Set the two expressions equal to each other:**
   \[
   3a + 5 = 4b + 2
   \]

4. **Solve for \(a\) in terms of \(b\):**
   \[
   3a + 5 = 4b + 2 \implies 3a = 4b - 3 \implies a = \frac{4b - 3}{3}
   \]

5. **Find integer solutions for \(a\) and \(b\):**
   - \(a\) must be an integer, so \(4b - 3\) must be divisible by 3.
   - Check values of \(b\) starting from the smallest possible integer greater than or equal to 5 (since \(42_b\) implies \(b \geq 5\)).

6. **Check \(b = 5\):**
   \[
   a = \frac{4(5) - 3}{3} = \frac{20 - 3}{3} = \frac{17}{3} \quad \text{(not an integer)}
   \]

7. **Check \(b = 6\):**
   \[
   a = \frac{4(6) - 3}{3} = \frac{24 - 3}{3} = \frac{21}{3} = 7 \quad \text{(integer)}
   \]

8. **Verify the solution:**
   - For \(a = 7\) and \(b = 6\):
     \[
     35_7 = 3(7) + 5 = 21 + 5 = 26
     \]
     \[
     42_6 = 4(6) + 2 = 24 + 2 = 26
     \]
   - Both values are equal, confirming the solution.

9. **Calculate the sum \(a + b\):**
   \[
   a + b = 7 + 6 = 13
   \]

The final answer is \(\boxed{13}\)