Difference between revisions of "2022 AMC 10A Problems"
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==Problem 8== | ==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 average (arithmetic mean) of the <math>6</math> numbers equals a value in the data set. What is the sum of all | + | 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>? |
<math>\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40</math> | <math>\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40</math> | ||
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==Problem 10== | ==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 <math>4\sqrt{2}</math> centimeters, as shown below. What is the area of the original index card? | + | 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? | ||
<asy> | <asy> | ||
− | / | + | // Diagram by MRENTHUSIASM, edited by Djmathman |
− | size( | + | 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)); | |
− | draw(( | + | label("$1$",(450,-80)); |
− | + | label("$1$",(475,-106)); | |
− | + | label("$8$",(300,-103)); | |
− | + | label("$4\sqrt 2$",(300,-173)); | |
− | |||
− | draw(( | ||
− | draw(( | ||
− | label("$1$",( | ||
− | label("$1$",( | ||
− | label("$8$", | ||
− | label("$4\sqrt | ||
</asy> | </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> | <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> | ||
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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 | 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> | <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> | + | (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) } 9 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14</math> | <math>\textbf{(A) } 9 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14</math> | ||
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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 | 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}\ | + | <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>? | What is the remainder when <math>h</math> is divided by <math>17</math>? | ||
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==Problem 25== | ==Problem 25== | ||
− | 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 x-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> | + | 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> | <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$", (55/2,55/2)); | |
− | filldraw((0,0)--( | + | filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white); |
− | filldraw((0,0)--(0, | + | label(scale(1.3)*"$S$",(-14,14)); |
− | filldraw((- | + | filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white); |
− | + | label(scale(1.3)*"$T$",(3.5,25/2)); | |
− | + | draw((0,-10)--(0,60),EndArrow()); | |
− | draw((0,- | + | draw((-34,0)--(60,0),EndArrow()); |
− | |||
− | |||
− | |||
− | |||
</asy> | </asy> | ||
− | The fraction of lattice points in <math>S</math> that are in <math>S \cap T</math> is 27 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>? | + | 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) }336\qquad\textbf{(B) }337\qquad\textbf{(C) }338\qquad\textbf{(D) }339\qquad\textbf{(E) }340</math> | <math>\textbf{(A) }336\qquad\textbf{(B) }337\qquad\textbf{(C) }338\qquad\textbf{(D) }339\qquad\textbf{(E) }340</math> | ||
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==See also== | ==See also== | ||
− | {{AMC10 box|year= | + | {{AMC10 box|year=2022|ab=A|before=[[2021 Fall AMC 10B Problems]]|after=[[2022 AMC 10B Problems]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 17:45, 5 August 2024
2022 AMC 10A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
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 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
What is the value of
Problem 2
Mike cycled laps in
minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first
minutes?
Problem 3
The sum of three numbers is The first number is
times the third number, and the third number is
less than the second number. What is the absolute value of the difference between the first and second numbers?
Problem 4
In some countries, automobile fuel efficiency is measured in liters per kilometers while other countries use miles per gallon. Suppose that 1 kilometer equals
miles, and
gallon equals
liters. Which of the following gives the fuel efficiency in liters per
kilometers for a car that gets
miles per gallon?
Problem 5
Square has side length
. Points
,
,
, and
each lie on a side of
such that
is an equilateral convex hexagon with side length
. What is
?
Problem 6
Which expression is equal to for
Problem 7
The least common multiple of a positive integer and
is
, and the greatest common divisor of
and
is
. What is the sum of the digits of
?
Problem 8
A data set consists of (not distinct) positive integers:
,
,
,
,
, and
. The average (arithmetic mean) of the
numbers equals a value in the data set. What is the sum of all possible values of
?
Problem 9
A rectangle is partitioned into 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?
Problem 10
Daniel finds a rectangular index card and measures its diagonal to be centimeters.
Daniel then cuts out equal squares of side
cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be
centimeters, as shown below. What is the area of the original index card?
Problem 11
Ted mistakenly wrote as
What is the sum of all real numbers
for which these two expressions have the same value?
Problem 12
On Halloween 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
children who answered yes.
"Are you an alternater?" The principal gave a piece of candy to each of the
children who answered yes.
"Are you a liar?" The principal gave a piece of candy to each of the children who
answered yes.
How many pieces of candy in all did the principal give to the children who always tell the truth?
Problem 13
Let be a scalene triangle. Point
lies on
so that
bisects
The line through
perpendicular to
intersects the line through
parallel to
at point
Suppose
and
What is
Problem 14
How many ways are there to split the integers through
into
pairs such that in each pair, the greater number is at least
times the lesser number?
Problem 15
Quadrilateral with side lengths
is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form
where
and
are positive integers such that
and
have no common prime factor. What is
Problem 16
The roots of the polynomial 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
units. What is the volume of the new box?
Problem 17
How many three-digit positive integers are there whose nonzero digits
and
satisfy
(The bar indicates repetition, thus
is the infinite repeating decimal
)
Problem 18
Let be the transformation of the coordinate plane that first rotates the plane
degrees counterclockwise around the origin and then reflects the plane across the
-axis. What is the least positive integer
such that performing the sequence of transformations
returns the point
back to itself?
Problem 19
Define as the least common multiple of all the integers from
to
inclusive. There is a unique integer
such that
What is the remainder when
is divided by
?
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 ,
, and
. What is the fourth term of this sequence?
Problem 21
A bowl is formed by attaching four regular hexagons of side to a square of side
. 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?
Problem 22
Suppose that cards numbered
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
are picked up on the first pass,
and
on the second pass,
on the third pass,
on the fourth pass, and
on the fifth pass. For how many of the
possible orderings of the cards will the
cards be picked up in exactly two passes?
Problem 23
Isosceles trapezoid has parallel sides
and
with
and
There is a point
in the plane such that
and
What is
Problem 24
How many strings of length formed from the digits
,
,
,
,
are there such that for each
, at least
of the digits are less than
? (For example,
satisfies this condition
because it contains at least
digit less than
, at least
digits less than
, at least
digits less
than
, and at least
digits less than
. The string
does not satisfy the condition because it
does not contain at least
digits less than
.)
Problem 25
Let ,
, and
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
-axis. The left edge of
and the right edge of
are on the
-axis, and
contains
as many lattice points as does
. The top two vertices of
are in
, and
contains
of the lattice points contained in
See the figure (not drawn to scale).
The fraction of lattice points in
that are in
is
times the fraction of lattice points in
that are in
. What is the minimum possible value of the edge length of
plus the edge length of
plus the edge length of
?
See also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2021 Fall AMC 10B Problems |
Followed by 2022 AMC 10B Problems | |
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 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.