Difference between revisions of "2022 AMC 10A Problems"

Latest revision as of 18:03, 3 November 2024

2022 AMC 10A (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator). Figures are not necessarily drawn to scale. You will have 75 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

Problem 1

What is the value of $3+\frac{1}{3+\frac{1}{3+\frac13}}?$ $\textbf{(A)}\ \frac{31}{10}\qquad\textbf{(B)}\ \frac{49}{15}\qquad\textbf{(C)}\ \frac{33}{10}\qquad\textbf{(D)}\ \frac{109}{33}\qquad\textbf{(E)}\ \frac{15}{4}$

Solution

Problem 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?

$\textbf{(A) } 5 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 9 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 13$

Solution

Problem 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?

$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$

Solution

Problem 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?

$\textbf{(A) } \frac{x}{100lm} \qquad \textbf{(B) } \frac{xlm}{100} \qquad \textbf{(C) } \frac{lm}{100x} \qquad \textbf{(D) } \frac{100}{xlm} \qquad \textbf{(E) } \frac{100lm}{x}$

Solution

Problem 5

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 $s$ ?

$\textbf{(A) } \frac{\sqrt{2}}{3} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } 2 - \sqrt{2} \qquad \textbf{(D) } 1 - \frac{\sqrt{2}}{4} \qquad \textbf{(E) } \frac{2}{3}$

Solution

Problem 6

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

$\textbf{(A) } 3-2a \qquad \textbf{(B) } 1-a \qquad \textbf{(C) } 1 \qquad \textbf{(D) } a+1 \qquad \textbf{(E) } 3$

Solution

Problem 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 $n$ ?

$\textbf{(A) } 3 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 12$

Solution

Problem 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 possible values of $X$ ?

$\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40$

Solution

Problem 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?

$[asy] size(5.5cm); draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((2,0)--(8,0)--(8,2)--(2,2)--cycle); draw((8,0)--(12,0)--(12,2)--(8,2)--cycle); draw((0,2)--(6,2)--(6,4)--(0,4)--cycle); draw((6,2)--(12,2)--(12,4)--(6,4)--cycle); [/asy]$

$\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720$

Solution

Problem 10

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, as shown below. What is the area of the original index card? $[asy] // Diagram by MRENTHUSIASM, edited by Djmathman size(200); defaultpen(linewidth(0.6)); draw((489.5,-213) -- (225.5,-213) -- (225.5,-185) -- (199.5,-185) -- (198.5,-62) -- (457.5,-62) -- (457.5,-93) -- (489.5,-93) -- cycle); draw((206.29,-70.89) -- (480.21,-207.11), linetype ("6 6"),Arrows(size=4,arrowhead=HookHead)); draw((237.85,-182.24) -- (448.65,-95.76),linetype ("6 6"),Arrows(size=4,arrowhead=HookHead)); label("$1$",(450,-80)); label("$1$",(475,-106)); label("$8$",(300,-103)); label("$4\sqrt 2$",(300,-173)); [/asy]$ $\textbf{(A) } 14 \qquad \textbf{(B) } 10\sqrt{2} \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 12\sqrt{2} \qquad \textbf{(E) } 18$

Solution

Problem 11

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

$\textbf{(A) } 5 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$

Solution

Problem 12

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?

$\textbf{(A) } 7 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 21 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 31$

Solution

Problem 13

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 $AD?$

$\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 12$

Solution

Problem 14

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?

$\textbf{(A) } 108 \qquad \textbf{(B) } 120 \qquad \textbf{(C) } 126 \qquad \textbf{(D) } 132 \qquad \textbf{(E) } 144$

Solution

Problem 15

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 $\frac{a\pi-b}{c},$ where $a,b,$ and $c$ are positive integers such that $a$ and $c$ have no common prime factor. What is $a+b+c?$

$\textbf{(A) } 260 \qquad \textbf{(B) } 855 \qquad \textbf{(C) } 1235 \qquad \textbf{(D) } 1565 \qquad \textbf{(E) } 1997$

Solution

Problem 16

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?

$\textbf{(A) } \frac{24}{5} \qquad \textbf{(B) } \frac{42}{5} \qquad \textbf{(C) } \frac{81}{5} \qquad \textbf{(D) } 30 \qquad \textbf{(E) } 48$

Solution

Problem 17

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

$\textbf{(A) } 9 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14$

Solution

Problem 18

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 $T_1, T_2, T_3,...,T_n$ returns the point $(1, 0)$ back to itself?

$\textbf{(A) } 359 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721$

Solution

Problem 19

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 $17$ ?

$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$

Solution

Problem 20

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?

$\textbf{(A) } 190 \qquad \textbf{(B) } 194 \qquad \textbf{(C) } 198 \qquad \textbf{(D) } 202 \qquad \textbf{(E) } 206$

Solution

Problem 21

A bowl is formed by attaching four regular hexagons of side $1$ to a square of side $1$ . The edges of the adjacent hexagons coincide, as shown in the figure. 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? $[asy] import three; size(225); currentprojection= orthographic(camera=(-5.52541796301147,-2.61548797564715,1.6545450372312), up=(0.00247902062334861,0.000877141782387748,0.00966536329192992), target=(0,0,0), zoom=0.570588560870951); currentpen = black+1.5bp; triple A = O; triple M = (X+Y)/2; triple B = (-1/2,-1/2,1/sqrt(2)); triple C = (-1,0,sqrt(2)); triple D = (0,-1,sqrt(2)); transform3 rho = rotate(90,M,M+Z); //arrays of vertices for the lower level (the square), the middle level, //and the interleaves vertices of the upper level (the octagon) triple[] lVs = {A}; triple[] mVs = {B}; triple[] uVsl = {C}; triple[] uVsr = {D}; for(int i = 0; i < 3; ++i){ lVs.push(rho*lVs[i]); mVs.push(rho*mVs[i]); uVsl.push(rho*uVsl[i]); uVsr.push(rho*uVsr[i]); } lVs.cyclic = true; uVsl.cyclic = true; for(int i : new int[] {0,1,2,3}){ draw(uVsl[i]--uVsr[i]); draw(uVsr[i]--uVsl[i+1]); } draw(lVs[0]--lVs[1]^^lVs[0]--lVs[3]); for(int i : new int[] {0,1,3}){ draw(lVs[0]--lVs[i]); draw(lVs[i]--mVs[i]); draw(mVs[i]--uVsl[i]); } for(int i : new int[] {0,3}){ draw(mVs[i]--uVsr[i]); } for(int i : new int[] {1,3}) draw(lVs[2]--lVs[i],dashed); draw(lVs[2]--mVs[2],dashed); draw(mVs[2]--uVsl[2]^^mVs[2]--uVsr[2],dashed); draw(mVs[1]--uVsr[1],dashed); //Comment two lines below to remove red edges //draw(lVs[1]--lVs[3],red+2bp); //draw(uVsl[0]--uVsr[0],red+2bp); [/asy]$ $\textbf{(A) } 6 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 5+2\sqrt{2} \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$

Solution

Problem 22

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?

$[asy] size(11cm); draw((0,0)--(2,0)--(2,3)--(0,3)--cycle); label("7", (1,1.5)); draw((3,0)--(5,0)--(5,3)--(3,3)--cycle); label("11", (4,1.5)); draw((6,0)--(8,0)--(8,3)--(6,3)--cycle); label("8", (7,1.5)); draw((9,0)--(11,0)--(11,3)--(9,3)--cycle); label("6", (10,1.5)); draw((12,0)--(14,0)--(14,3)--(12,3)--cycle); label("4", (13,1.5)); draw((15,0)--(17,0)--(17,3)--(15,3)--cycle); label("5", (16,1.5)); draw((18,0)--(20,0)--(20,3)--(18,3)--cycle); label("9", (19,1.5)); draw((21,0)--(23,0)--(23,3)--(21,3)--cycle); label("12", (22,1.5)); draw((24,0)--(26,0)--(26,3)--(24,3)--cycle); label("1", (25,1.5)); draw((27,0)--(29,0)--(29,3)--(27,3)--cycle); label("13", (28,1.5)); draw((30,0)--(32,0)--(32,3)--(30,3)--cycle); label("10", (31,1.5)); draw((33,0)--(35,0)--(35,3)--(33,3)--cycle); label("2", (34,1.5)); draw((36,0)--(38,0)--(38,3)--(36,3)--cycle); label("3", (37,1.5)); [/asy]$ $\textbf{(A) } 4082 \qquad \textbf{(B) } 4095 \qquad \textbf{(C) } 4096 \qquad \textbf{(D) } 8178 \qquad \textbf{(E) } 8191$

Solution

Problem 23

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 $\tfrac{BC}{AD}?$

$\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{2}{3}\qquad\textbf{(E) }\frac{3}{4}$

Solution

Problem 24

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 $2$ .)

$\textbf{(A) }500\qquad\textbf{(B) }625\qquad\textbf{(C) }1089\qquad\textbf{(D) }1199\qquad\textbf{(E) }1296$

Solution

Problem 25

Let $R$ , $S$ , and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) 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 $\frac{9}{4}$ as many lattice points as does $S$ . The top two vertices of $T$ are in $R \cup S$ , and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S.$ See the figure (not drawn to scale). $[asy] size(8cm); label(scale(.8)*"$y$", (0,60), N); label(scale(.8)*"$x$", (60,0), E); filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white); label(scale(1.3)*"$R$", (55/2,55/2)); filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white); label(scale(1.3)*"$S$",(-14,14)); filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white); label(scale(1.3)*"$T$",(3.5,25/2)); draw((0,-10)--(0,60),EndArrow()); draw((-34,0)--(60,0),EndArrow()); [/asy]$ 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 $T$ ?

$\textbf{(A) }336\qquad\textbf{(B) }337\qquad\textbf{(C) }338\qquad\textbf{(D) }339\qquad\textbf{(E) }340$

Solution

2022 AMC 10A (Problems • Answer Key • Resources)
Preceded by 2021 Fall AMC 10B Problems	Followed by 2022 AMC 10B Problems
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Difference between revisions of "2022 AMC 10A Problems"

Latest revision as of 18:03, 3 November 2024

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also

@@ Line 50: / Line 50: @@
 ==Problem 7==
-The least common multiple of a positive divisor <math>n</math> and <math>18</math> is <math>180</math>, and the greatest common divisor of <math>n</math> and <math>45</math> is <math>15</math>. What is the sum of the digits of <math>n</math>?
+The least common multiple of a positive integer <math>n</math> and <math>18</math> is <math>180</math>, and the greatest common divisor of <math>n</math> and <math>45</math> is <math>15</math>. What is the sum of the digits of <math>n</math>?
 <math>\textbf{(A) } 3 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 12</math>
@@ Line 58: / Line 58: @@
 ==Problem 8==
-A data set consists of <math>6</math> not distinct) positive integers: <math>1</math>, <math>7</math>, <math>5</math>, <math>2</math>, <math>5</math>, and <math>X</math>. The
+A data set consists of <math>6</math> (not distinct) positive integers: <math>1</math>, <math>7</math>, <math>5</math>, <math>2</math>, <math>5</math>, and <math>X</math>. The average (arithmetic mean) of the <math>6</math> numbers equals a value in the data set. What is the sum of all possible values of <math>X</math>?
-average (arithmetic mean) of the <math>6</math> numbers equals a value in the data set. What is
-the sum of all positive values of <math>X</math>?
 <math>\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40</math>
@@ Line 68: / Line 66: @@
 ==Problem 9==
-XXX
+A rectangle is partitioned into <math>5</math> 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?
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+<asy> size(5.5cm); draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((2,0)--(8,0)--(8,2)--(2,2)--cycle); draw((8,0)--(12,0)--(12,2)--(8,2)--cycle); draw((0,2)--(6,2)--(6,4)--(0,4)--cycle); draw((6,2)--(12,2)--(12,4)--(6,4)--cycle); </asy>
+<math>\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720</math>
 [[2022 AMC 10A Problems/Problem 9|Solution]]
@@ Line 76: / Line 76: @@
 ==Problem 10==
-XXX
+Daniel finds a rectangular index card and measures its diagonal to be <math>8</math> centimeters.
+Daniel then cuts out equal squares of side <math>1</math> cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be <math>4\sqrt{2}</math> centimeters, as shown below. What is the area of the original index card?
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+<asy>
+// Diagram by MRENTHUSIASM, edited by Djmathman
+size(200);
+defaultpen(linewidth(0.6));
+draw((489.5,-213) -- (225.5,-213) -- (225.5,-185) -- (199.5,-185) -- (198.5,-62) -- (457.5,-62) -- (457.5,-93) -- (489.5,-93) -- cycle);
+draw((206.29,-70.89) -- (480.21,-207.11), linetype ("6 6"),Arrows(size=4,arrowhead=HookHead));
+draw((237.85,-182.24) -- (448.65,-95.76),linetype ("6 6"),Arrows(size=4,arrowhead=HookHead));
+label("$1$",(450,-80));
+label("$1$",(475,-106));
+label("$8$",(300,-103));
+label("$4\sqrt 2$",(300,-173));
+</asy>
+<math>\textbf{(A) } 14 \qquad \textbf{(B) } 10\sqrt{2} \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 12\sqrt{2} \qquad \textbf{(E) } 18</math>
 [[2022 AMC 10A Problems/Problem 10|Solution]]
@@ Line 84: / Line 96: @@
 ==Problem 11==
-XXX
+Ted mistakenly wrote <math>2^m\cdot\sqrt{\frac{1}{4096}}</math> as <math>2\cdot\sqrt[m]{\frac{1}{4096}}.</math> What is the sum of all real numbers <math>m</math> for which these two expressions have the same value?
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+<math>\textbf{(A) } 5 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9</math>
 [[2022 AMC 10A Problems/Problem 11|Solution]]
@@ Line 92: / Line 104: @@
 ==Problem 12==
-XXX
+On Halloween, <math>31</math> 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 <math>22</math>
+children who answered yes.
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+"Are you an alternater?" The principal gave a piece of candy to each of the <math>15</math>
+children who answered yes.
+"Are you a liar?" The principal gave a piece of candy to each of the <math>9</math> children who
+answered yes.
+How many pieces of candy in all did the principal give to the children who always
+tell the truth?
+<math>\textbf{(A) } 7 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 21 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 31</math>
 [[2022 AMC 10A Problems/Problem 12|Solution]]
@@ Line 100: / Line 129: @@
 ==Problem 13==
-XXX
+Let <math>\triangle ABC</math> be a scalene triangle. Point <math>P</math> lies on <math>\overline{BC}</math> so that <math>\overline{AP}</math> bisects <math>\angle BAC.</math> The line through <math>B</math> perpendicular to <math>\overline{AP}</math> intersects the line through <math>A</math> parallel to <math>\overline{BC}</math> at point <math>D.</math> Suppose <math>BP=2</math> and <math>PC=3.</math> What is <math>AD?</math>
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+<math>\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 12</math>
 [[2022 AMC 10A Problems/Problem 13|Solution]]
@@ Line 108: / Line 137: @@
 ==Problem 14==
-XXX
+How many ways are there to split the integers <math>1</math> through <math>14</math> into <math>7</math> pairs such that in each pair, the greater number is at least <math>2</math> times the lesser number?
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+<math>\textbf{(A) } 108 \qquad \textbf{(B) } 120 \qquad \textbf{(C) } 126 \qquad \textbf{(D) } 132 \qquad \textbf{(E) } 144</math>
 [[2022 AMC 10A Problems/Problem 14|Solution]]
@@ Line 116: / Line 145: @@
 ==Problem 15==
-XXX
+Quadrilateral <math>ABCD</math> with side lengths <math>AB=7, BC=24, CD=20, DA=15</math> is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form <math>\frac{a\pi-b}{c},</math> where <math>a,b,</math> and <math>c</math> are positive integers such that <math>a</math> and <math>c</math> have no common prime factor. What is <math>a+b+c?</math>
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+<math>\textbf{(A) } 260 \qquad \textbf{(B) } 855 \qquad \textbf{(C) } 1235 \qquad \textbf{(D) } 1565 \qquad \textbf{(E) } 1997</math>
 [[2022 AMC 10A Problems/Problem 15|Solution]]
@@ Line 124: / Line 153: @@
 ==Problem 16==
-XXX
+The roots of the polynomial <math>10x^3 - 39x^2 + 29x - 6</math> 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 <math>2</math> units. What is the volume of the new box?
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+<math>\textbf{(A) } \frac{24}{5} \qquad \textbf{(B) } \frac{42}{5} \qquad \textbf{(C) } \frac{81}{5} \qquad \textbf{(D) } 30 \qquad \textbf{(E) } 48</math>
 [[2022 AMC 10A Problems/Problem 16|Solution]]
@@ Line 132: / Line 161: @@
 ==Problem 17==
-XXX
+How many three-digit positive integers <math>\underline{a} \ \underline{b} \ \underline{c}</math> are there whose nonzero digits <math>a,b,</math> and <math>c</math> satisfy
+<cmath>0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?</cmath>
+(The bar indicates repetition, thus <math>0.\overline{\underline{a}~\underline{b}~\underline{c}}</math> is the infinite repeating decimal <math>0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots</math>)
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+<math>\textbf{(A) } 9 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14</math>
 [[2022 AMC 10A Problems/Problem 17|Solution]]
 ==Problem 18==
+Let <math>T_k</math> be the transformation of the coordinate plane that first rotates the plane <math>k</math> degrees counterclockwise around the origin and then reflects the plane across the <math>y</math>-axis. What is the least positive integer <math>n</math> such that performing the sequence of transformations <math>T_1, T_2, T_3,...,T_n</math> returns the point <math>(1, 0)</math> back to itself?
-XXX
+<math>\textbf{(A) } 359 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721 </math>
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
 [[2022 AMC 10A Problems/Problem 18|Solution]]
@@ Line 148: / Line 178: @@
 ==Problem 19==
-XXX
+Define <math>L_n</math> as the least common multiple of all the integers from <math>1</math> to <math>n</math> inclusive. There is a unique integer <math>h</math> such that
+<cmath>\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{17}=\frac{h}{L_{17}}</cmath>
+What is the remainder when <math>h</math> is divided by <math>17</math>?
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+<math>\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9</math>
 [[2022 AMC 10A Problems/Problem 19|Solution]]
@@ Line 156: / Line 188: @@
 ==Problem 20==
-XXX
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+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 <math>57</math>, <math>60</math>, and <math>91</math>. What is the fourth term of this sequence?
+<math>\textbf{(A) } 190 \qquad \textbf{(B) } 194 \qquad \textbf{(C) } 198 \qquad \textbf{(D) } 202 \qquad \textbf{(E) } 206</math>
 [[2022 AMC 10A Problems/Problem 20|Solution]]
@@ Line 164: / Line 197: @@
 ==Problem 21==
-XXX
+A bowl is formed by attaching four regular hexagons of side <math>1</math> to a square of side <math>1</math>. The edges of the adjacent hexagons coincide, as shown in the figure. 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?
+<asy>
+import three;
+size(225);
+currentprojection=
+  orthographic(camera=(-5.52541796301147,-2.61548797564715,1.6545450372312),
+               up=(0.00247902062334861,0.000877141782387748,0.00966536329192992),
+               target=(0,0,0),
+               zoom=0.570588560870951);
+currentpen = black+1.5bp;
+triple A = O;
+triple M = (X+Y)/2;
+triple B = (-1/2,-1/2,1/sqrt(2));
+triple C = (-1,0,sqrt(2));
+triple D = (0,-1,sqrt(2));
+transform3 rho = rotate(90,M,M+Z);
+//arrays of vertices for the lower level (the square), the middle level,
+//and the interleaves vertices of the upper level (the octagon)
+triple[] lVs = {A};
+triple[] mVs = {B};
+triple[] uVsl = {C};
+triple[] uVsr = {D};
+for(int i = 0; i < 3; ++i){
+  lVs.push(rho*lVs[i]);
+  mVs.push(rho*mVs[i]);
+  uVsl.push(rho*uVsl[i]);
+  uVsr.push(rho*uVsr[i]);
+}
+lVs.cyclic = true;
+uVsl.cyclic = true;
+for(int i : new int[] {0,1,2,3}){
+  draw(uVsl[i]--uVsr[i]);
+  draw(uVsr[i]--uVsl[i+1]);
+}
+draw(lVs[0]--lVs[1]^^lVs[0]--lVs[3]);
+for(int i : new int[] {0,1,3}){
+  draw(lVs[0]--lVs[i]);
+  draw(lVs[i]--mVs[i]);
+  draw(mVs[i]--uVsl[i]);
+}
+for(int i : new int[] {0,3}){
+  draw(mVs[i]--uVsr[i]);
+}
+for(int i : new int[] {1,3}) draw(lVs[2]--lVs[i],dashed);
+draw(lVs[2]--mVs[2],dashed);
+draw(mVs[2]--uVsl[2]^^mVs[2]--uVsr[2],dashed);
+draw(mVs[1]--uVsr[1],dashed);
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+//Comment two lines below to remove red edges
+//draw(lVs[1]--lVs[3],red+2bp);
+//draw(uVsl[0]--uVsr[0],red+2bp);
+</asy>
+<math>\textbf{(A) } 6 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 5+2\sqrt{2} \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9</math>
 [[2022 AMC 10A Problems/Problem 21|Solution]]
@@ Line 172: / Line 260: @@
 ==Problem 22==
-XXX
+Suppose that <math>13</math> cards numbered <math>1, 2, 3, \ldots, 13</math> 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 <math>1, 2, 3</math> are picked up on the first pass, <math>4</math> and <math>5</math> on the second pass, <math>6</math> on the third pass, <math>7, 8, 9, 10</math> on the fourth pass, and <math>11, 12, 13</math> on the fifth pass. For how many of the <math>13!</math> possible orderings of the cards will the <math>13</math> cards be picked up in exactly two passes?
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+<asy>
+size(11cm);
+draw((0,0)--(2,0)--(2,3)--(0,3)--cycle);
+label("7", (1,1.5));
+draw((3,0)--(5,0)--(5,3)--(3,3)--cycle);
+label("11", (4,1.5));
+draw((6,0)--(8,0)--(8,3)--(6,3)--cycle);
+label("8", (7,1.5));
+draw((9,0)--(11,0)--(11,3)--(9,3)--cycle);
+label("6", (10,1.5));
+draw((12,0)--(14,0)--(14,3)--(12,3)--cycle);
+label("4", (13,1.5));
+draw((15,0)--(17,0)--(17,3)--(15,3)--cycle);
+label("5", (16,1.5));
+draw((18,0)--(20,0)--(20,3)--(18,3)--cycle);
+label("9", (19,1.5));
+draw((21,0)--(23,0)--(23,3)--(21,3)--cycle);
+label("12", (22,1.5));
+draw((24,0)--(26,0)--(26,3)--(24,3)--cycle);
+label("1", (25,1.5));
+draw((27,0)--(29,0)--(29,3)--(27,3)--cycle);
+label("13", (28,1.5));
+draw((30,0)--(32,0)--(32,3)--(30,3)--cycle);
+label("10", (31,1.5));
+draw((33,0)--(35,0)--(35,3)--(33,3)--cycle);
+label("2", (34,1.5));
+draw((36,0)--(38,0)--(38,3)--(36,3)--cycle);
+label("3", (37,1.5));
+</asy>
+<math>\textbf{(A) } 4082 \qquad \textbf{(B) } 4095 \qquad \textbf{(C) } 4096 \qquad \textbf{(D) } 8178 \qquad \textbf{(E) } 8191</math>
 [[2022 AMC 10A Problems/Problem 22|Solution]]
@@ Line 180: / Line 297: @@
 ==Problem 23==
-XXX
+Isosceles trapezoid <math>ABCD</math> has parallel sides <math>\overline{AD}</math> and <math>\overline{BC},</math> with <math>BC < AD</math> and <math>AB = CD.</math> There is a point <math>P</math> in the plane such that <math>PA=1, PB=2, PC=3,</math> and <math>PD=4.</math> What is <math>\tfrac{BC}{AD}?</math>
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+<math>\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{2}{3}\qquad\textbf{(E) }\frac{3}{4}</math>
 [[2022 AMC 10A Problems/Problem 23|Solution]]
@@ Line 188: / Line 305: @@
 ==Problem 24==
-XXX
+How many strings of length <math>5</math> formed from the digits <math>0</math>, <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math> are there such that for each <math>j \in \{1,2,3,4\}</math>, at least <math>j</math> of the digits are less than <math>j</math>? (For example, <math>02214</math> satisfies this condition
+because it contains at least <math>1</math> digit less than <math>1</math>, at least <math>2</math> digits less than <math>2</math>, at least <math>3</math> digits less
+than <math>3</math>, and at least <math>4</math> digits less than <math>4</math>. The string <math>23404</math> does not satisfy the condition because it
+does not contain at least <math>2</math> digits less than <math>2</math>.)
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+<math>\textbf{(A) }500\qquad\textbf{(B) }625\qquad\textbf{(C) }1089\qquad\textbf{(D) }1199\qquad\textbf{(E) }1296</math>
 [[2022 AMC 10A Problems/Problem 24|Solution]]
@@ Line 196: / Line 316: @@
 ==Problem 25==
-XXX
+Let <math>R</math>, <math>S</math>, and <math>T</math> be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the <math>x</math>-axis. The left edge of <math>R</math> and the right edge of <math>S</math> are on the <math>y</math>-axis, and <math>R</math> contains <math>\frac{9}{4}</math> as many lattice points as does <math>S</math>. The top two vertices of <math>T</math> are in <math>R \cup S</math>, and <math>T</math> contains <math>\frac{1}{4}</math> of the lattice points contained in <math>R \cup S.</math> See the figure (not drawn to scale).
+<asy>
+size(8cm);
+label(scale(.8)*"$y$", (0,60), N);
+label(scale(.8)*"$x$", (60,0), E);
+filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white);
+label(scale(1.3)*"$R$", (55/2,55/2));
+filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white);
+label(scale(1.3)*"$S$",(-14,14));
+filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white);
+label(scale(1.3)*"$T$",(3.5,25/2));
+draw((0,-10)--(0,60),EndArrow());
+draw((-34,0)--(60,0),EndArrow());
+</asy>
+The fraction of lattice points in <math>S</math> that are in <math>S \cap T</math> is <math>27</math> times the fraction of lattice points in <math>R</math> that are in <math>R \cap T</math>. What is the minimum possible value of the edge length of <math>R</math> plus the edge length of <math>S</math> plus the edge length of <math>T</math>?
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+<math>\textbf{(A) }336\qquad\textbf{(B) }337\qquad\textbf{(C) }338\qquad\textbf{(D) }339\qquad\textbf{(E) }340</math>
 [[2022 AMC 10A Problems/Problem 25|Solution]]
 ==See also==
-{{AMC10 box|year=2020|ab=A|before=[[2021 Fall AMC 10B Problems]]|after=[[2022 AMC 10B Problems]]}}
+{{AMC10 box|year=2022|ab=A|before=[[2021 Fall AMC 10B Problems]]|after=[[2022 AMC 10B Problems]]}}
 {{MAA Notice}}