AMC 10A 2022

What is the value of $3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}?$

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?

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?

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?

Square $ABCD$ has side length $1$. Points $P$, $Q$, $R$, and $S$ each lie on a side of $ABCD$ such that $APQCRS$ is an equilateral convex hexagon with side length $s$. What is $s$?

Which expression is equal to $\left|a-2-\sqrt{(a-1)^2}\right|$ for $a < 0$?

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 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 possible values of $X$?

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?

The figure shows a $3 \times 2$ grid of rectangles where: the bottom row has three side-by-side regions (left, middle, right), and the top row has two regions split differently (the left top region spans above regions 1 and part of 2; the right top region spans the rest). Adjacencies: region 1 touches 2 and 4; region 2 touches 1, 3, 4, and 5; region 3 touches 2 and 5; region 4 touches 1, 2, and 5; region 5 touches 2, 3, and 4.

Daniel finds a rectangular index card and measures its diagonal to be $8$ centimeters. Daniel then cuts out equal squares of side $1$ 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. What is the area of the original index card?

Ted mistakenly wrote $2^m \cdot \sqrt{\dfrac{1}{4096}}$ as $2 \cdot \sqrt[m]{\dfrac{1}{4096}}$. What is the sum of all real numbers $m$ for which these two expressions have the same value?

On Halloween, $31$ children walked into the principal's office asking for candy. They can be classified into three types: some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.

"Are you a truth-teller?" The principal gave a piece of candy to each of the $22$ children who answered yes.

"Are you an alternater?" The principal gave a piece of candy to each of the $15$ children who answered yes.

"Are you a liar?" The principal gave a piece of candy to each of the $9$ children who answered yes.

How many pieces of candy in all did the principal give to the children who always tell the truth?

Let $\triangle ABC$ be a scalene triangle. Point $P$ lies on $\overline{BC}$ so that $\overline{AP}$ bisects $\angle BAC.$ The line through $B$ perpendicular to $\overline{AP}$ intersects the line through $A$ parallel to $\overline{BC}$ at point $D.$ Suppose $BP = 2$ and $PC = 3.$ What is $AD$?

How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser number?

Quadrilateral $ABCD$ with side lengths $AB=7$, $BC=24$, $CD=20$, $DA=15$ is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form $\dfrac{a\pi - b}{c}$, where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ have no common prime factor. What is $a+b+c$?

The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?

How many three-digit positive integers $\underline{a}\ \underline{b}\ \underline{c}$ are there whose nonzero digits $a$, $b$, and $c$ satisfy $0.\overline{\underline{a}\ \underline{b}\ \underline{c}} = \frac{1}{3}\!\left(0.\overline{a} + 0.\overline{b} + 0.\overline{c}\right)?$ (The bar indicates repetition, so $0.\overline{\underline{a}\ \underline{b}\ \underline{c}}$ is the infinite repeating decimal $0.\underline{a}\ \underline{b}\ \underline{c}\ \underline{a}\ \underline{b}\ \underline{c}\ \cdots$)

Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \ldots, T_n$ returns the point $(1,0)$ back to itself?

Define $L_n$ as the least common multiple of all the integers from $1$ to $n$ inclusive. There is a unique integer $h$ such that $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{17}=\frac{h}{L_{17}}.$ What is the remainder when $h$ is divided by $17$?

A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are $57$, $60$, and $91$. What is the fourth term of this sequence?

A bowl is formed by attaching four regular hexagons of side $1$ to a square of side $1$. The edges of adjacent hexagons coincide. (The square forms the base and each hexagon folds up to form a side, creating a three-dimensional bowl shape.) What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?

Suppose that $13$ cards numbered $1, 2, 3, \ldots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards $1, 2, 3$ are picked up on the first pass, $4$ and $5$ on the second pass, $6$ on the third pass, $7, 8, 9, 10$ on the fourth pass, and $11, 12, 13$ on the fifth pass. For how many of the $13!$ possible orderings of the cards will the $13$ cards be picked up in exactly two passes?

Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC}$, with $BC < AD$ and $AB = CD$. There is a point $P$ in the plane such that $PA = 1$, $PB = 2$, $PC = 3$, and $PD = 4$. What is $\dfrac{BC}{AD}$?

How many strings of length $5$ formed from the digits $0, 1, 2, 3, 4$ are there such that for each $j \in \{1, 2, 3, 4\}$, at least $j$ of the digits are less than $j$? (For example, $02214$ satisfies this condition because it contains at least $1$ digit less than $1$, at least $2$ digits less than $2$, at least $3$ digits less than $3$, and at least $4$ digits less than $4$. The string $23404$ does not satisfy the condition because it does not contain at least $2$ digits less than $2$.)

Let $R$, $S$, and $T$ be squares that have vertices at lattice points in the coordinate plane, together with their interiors. The bottom edge of each square is on the $x$-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\dfrac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\dfrac{1}{4}$ of the lattice points contained in $R \cup S$. The fraction of lattice points in $S$ that are in $S \cap T$ is $27$ times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$?