1. What is the value of
$
3+\frac{1}{3+\frac{1}{3+\frac{1}{3}} ?}
$
(A) $\frac{31}{10}$
(B) $\frac{49}{15}$
(C) $\frac{33}{10}$
(D) $\frac{109}{33}$
(E) $\frac{15}{4}$
2. Mike cycled 15 laps in 57 minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first 27 minutes?
(A) 5
(B) 7
(C) 9
(D) 11
(E) 13
3. The sum of three numbers is 96. The first number is 6 times the third number, and the third number is 40 less than the second number. What is the absolute value of the difference between the first and second numbers?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
4. In some countries, automobile fuel efficiency is measured in liters per 100 kilometers while other countries use miles per gallon. Suppose that 1 kilometer equals $m$ miles, and 1 gallon equals $l$ liters. Which of the following gives the fuel efficiency in liters per 100 kilometers for a car that gets $x$ miles per gallon?
(A) $\frac{x}{100 l m}$
(B) $\frac{x l m}{100}$
(C) $\frac{l m}{100 x}$
(D) $\frac{100}{x l m}$
(E) $\frac{100 l m}{x}$
5. Square $A B C D$ has side length 1. Point $P, Q, R$, and $S$ each lie on a side of $A B C D$ such that $A P Q C R S$ is an equilateral convex hexagon with side length $s$. What is $s$?
(A) $\frac{\sqrt{2}}{3}$
(B) $\frac{1}{2}$
(C) $2-\sqrt{2}$
(D) $1-\frac{\sqrt{2}}{4}$
(E) $\frac{2}{3}$
6. Which expression is equal to
$
\left|a-2-\sqrt{(a-1)^2}\right|
$
for $a<0$ ?
(A) $3-2 a$
(B) $1-a$
(C) 1
(D) $a+1$
(E) 3
7. The least common multiple of a positive integer $n$ and 18 is 180 , and the greatest common divisor of $n$ and 45 is 15 . What is the sum of the digits of $n$ ?
(A) 3
(B) 6
(C) 8
(D) 9
(E) 12
8. A data set consists of 6 (not distinct) positive integers: $1,7,5,2,5$, and $X$. The average (arithmetic mean) of the 6 numbers equals a value in the data set. What is the sum of all positive values of $X$ ?
(A) 10
(B) 26
(C) 32
(D) 36
(E) 40
9. A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
![[amc10a2022p9.svg]]
(A) 120
(B) 270
(C) 360
(D) 540
(E) 720
10. Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side $1 \mathrm{~cm}$ at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4 \sqrt{2}$ centimeters, as shown below. What is the area of the original index card?