Difference between revisions of "2023 AMC 10A Problems"
(→Problem 4) |
Vincentli10 (talk | contribs) |
||
Line 2: | Line 2: | ||
==Problem 1== | ==Problem 1== | ||
+ | |||
Cities <math>A</math> and <math>B</math> are <math>45</math> miles apart. Alicia lives in <math>A</math> and Beth lives in <math>B</math>. Alicia bikes towards <math>B</math> at 18 miles per hour. Leaving at the same time, Beth bikes toward <math>A</math> at 12 miles per hour. How many miles from City <math>A</math> will they be when they meet? | Cities <math>A</math> and <math>B</math> are <math>45</math> miles apart. Alicia lives in <math>A</math> and Beth lives in <math>B</math>. Alicia bikes towards <math>B</math> at 18 miles per hour. Leaving at the same time, Beth bikes toward <math>A</math> at 12 miles per hour. How many miles from City <math>A</math> will they be when they meet? | ||
Line 25: | Line 26: | ||
A quadrilateral has all side lengths, a perimeter of <math>26</math>, and one side of length <math>4</math>. What is the greatest possible length of one side of this quadrilateral? | A quadrilateral has all side lengths, a perimeter of <math>26</math>, and one side of length <math>4</math>. What is the greatest possible length of one side of this quadrilateral? | ||
+ | |||
+ | [[2023 AMC 10A Problems/Problem 4|Solution]] | ||
<math>\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }12\qquad\textbf{(E) }13</math> | <math>\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }12\qquad\textbf{(E) }13</math> | ||
Line 39: | Line 42: | ||
==Problem 6== | ==Problem 6== | ||
− | An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is 21. What is the value of the cube? | + | |
+ | An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is <math>21</math>. What is the value of the cube? | ||
+ | |||
+ | <math>\textbf{(A) } 42 \qquad \textbf{(B) } 63 \qquad \textbf{(C) } 84 \qquad \textbf{(D) } 126 \qquad \textbf{(E) } 252</math> | ||
+ | |||
+ | [[2023 AMC 10A Problems/Problem 6|Solution]] | ||
+ | |||
+ | ==Problem 7== | ||
+ | |||
+ | Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3? | ||
+ | |||
+ | <math>\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{49}{216}\qquad\textbf{(C) }\frac{25}{108}\qquad\textbf{(D) }\frac{17}{72}\qquad\textbf{(E) }\frac{13}{54}</math> | ||
+ | |||
+ | ==Problem 8== | ||
+ | |||
+ | Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at <math>110</math> degrees Fahrenheit, which is <math>0</math> degrees on the Breadus scale. Bread is baked at <math>350</math> degrees Fahrenheit, which is <math>100</math> degrees on the Breadus scale. Bread is done when its internal temperature is <math>200</math> degrees Fahrenheit. What is this in degrees on the Breadus scale? | ||
+ | |||
+ | <math>\textbf{(A) }33\qquad\textbf{(B) }34.5\qquad\textbf{(C) }36\qquad\textbf{(D) }37.5\qquad\textbf{(E) }39</math> | ||
+ | |||
+ | ==Problem 9== | ||
+ | |||
+ | A digital display shows the current date as an <math>8</math>-digit integer consisting of a <math>4</math>-digit year, followed by a <math>2</math>-digit month, followed by a <math>2</math>-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in <math>2023</math> will each digit appear an even number of times in the 8-digital display for that date? | ||
+ | |||
+ | <math>\textbf{(A)}~5\qquad\textbf{(B)}~6\qquad\textbf{(C)}~7\qquad\textbf{(D)}~8\qquad\textbf{(E)}~9</math> | ||
+ | |||
+ | ==Problem 10== | ||
+ | |||
+ | Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an <math>11</math> on the next quiz, her mean will increase by <math>1</math>. If she scores an <math>11</math> on each of the next three quizzes, her mean will increase by <math>2</math>. What is the mean of her quiz scores currently? | ||
+ | |||
+ | <math>\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8</math> | ||
+ | |||
+ | ==Problem 11== | ||
+ | |||
+ | A square of area <math>2</math> is inscribed in a square of area <math>3</math>, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?[asy] size(200); defaultpen(linewidth(0.6pt)+fontsize(10pt)); real y = sqrt(3); pair A,B,C,D,E,F,G,H; A = (0,0); B = (0,y); C = (y,y); D = (y,0); E = ((y + 1)/2,y); F = (y, (y - 1)/2); G = ((y - 1)/2, 0); H = (0,(y + 1)/2); fill(H--B--E--cycle, gray); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); [/asy] | ||
+ | |||
+ | <math>\textbf{(A) }\frac15\qquad\textbf{(B) }\frac14\qquad\textbf{(C) }2-\sqrt3\qquad\textbf{(D) }\sqrt3-\sqrt2\qquad\textbf{(E) }\sqrt2-1</math> | ||
+ | |||
+ | ==Problem 12== | ||
+ | |||
+ | How many three-digit positive integers <math>N</math> satisfy the following properties? | ||
+ | |||
+ | The number <math>N</math> is divisible by <math>7</math>. | ||
+ | The number formed by reversing the digits of <math>N</math> is divisble by <math>5</math>. | ||
+ | |||
+ | <math>\textbf{(A) } 13 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 16 \qquad \textbf{(E) } 17</math> | ||
+ | |||
+ | ==Problem 13== | ||
+ | |||
+ | Abdul and Chiang are standing <math>48</math> feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chaing measures <math>60^\circ</math>. What is the square of the distance (in feet) between Abdul and Bharat? | ||
+ | |||
+ | <math>\textbf{(A) } 1728 \qquad \textbf{(B) } 2601 \qquad \textbf{(C) } 3072 \qquad \textbf{(D) } 4608 \qquad \textbf{(E) } 6912</math> | ||
+ | |||
+ | ==Problem 14== | ||
+ | |||
+ | A number is chosen at random from among the first <math>100</math> positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by <math>11</math>? | ||
+ | |||
+ | <math>\textbf{(A)}~\frac{4}{100}\qquad\textbf{(B)}~\frac{9}{200} \qquad \textbf{(C)}~\frac{1}{20} \qquad\textbf{(D)}~\frac{11}{200}\qquad\textbf{(E)}~\frac{3}{50}</math> | ||
+ | |||
+ | ==Problem 15== | ||
+ | |||
+ | An even number of circles are nested, starting with a radius of <math>1</math> and increasing by <math>1</math> each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius <math>2</math> but outside the circle of radius <math>1.</math> An example showing <math>8</math> circles is displayed below. What is the least number of circles needed to make the total shaded area at least <math>2023\pi</math>? | ||
+ | |||
+ | <math>\textbf{(A) } 46 \qquad \textbf{(B) } 48 \qquad \textbf{(C) } 56 \qquad \textbf{(D) } 60 \qquad \textbf{(E) } 64</math> | ||
==See also== | ==See also== | ||
{{AMC10 box|year=2023|ab=A|before=[[2022 Fall AMC 10B Problems]]|after=[[2023 AMC 10B Problems]]}} | {{AMC10 box|year=2023|ab=A|before=[[2022 Fall AMC 10B Problems]]|after=[[2023 AMC 10B Problems]]}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 21:59, 9 November 2023
2023 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
Problem 1
Cities and
are
miles apart. Alicia lives in
and Beth lives in
. Alicia bikes towards
at 18 miles per hour. Leaving at the same time, Beth bikes toward
at 12 miles per hour. How many miles from City
will they be when they meet?
Problem 2
The weight of of a large pizza together with
cups of orange slices is the same weight of
of a large pizza together with
cups of orange slices. A cup of orange slices weight
of a pound. What is the weight, in pounds, of a large pizza?
Problem 3
How many positive perfect squares less than are divisible by
?
Problem 4
A quadrilateral has all side lengths, a perimeter of , and one side of length
. What is the greatest possible length of one side of this quadrilateral?
Problem 5
How many digits are in the base-ten representation of ?
Problem 6
An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is . What is the value of the cube?
Problem 7
Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3?
Problem 8
Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at degrees Fahrenheit, which is
degrees on the Breadus scale. Bread is baked at
degrees Fahrenheit, which is
degrees on the Breadus scale. Bread is done when its internal temperature is
degrees Fahrenheit. What is this in degrees on the Breadus scale?
Problem 9
A digital display shows the current date as an -digit integer consisting of a
-digit year, followed by a
-digit month, followed by a
-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in
will each digit appear an even number of times in the 8-digital display for that date?
Problem 10
Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an on the next quiz, her mean will increase by
. If she scores an
on each of the next three quizzes, her mean will increase by
. What is the mean of her quiz scores currently?
Problem 11
A square of area is inscribed in a square of area
, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?[asy] size(200); defaultpen(linewidth(0.6pt)+fontsize(10pt)); real y = sqrt(3); pair A,B,C,D,E,F,G,H; A = (0,0); B = (0,y); C = (y,y); D = (y,0); E = ((y + 1)/2,y); F = (y, (y - 1)/2); G = ((y - 1)/2, 0); H = (0,(y + 1)/2); fill(H--B--E--cycle, gray); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); [/asy]
Problem 12
How many three-digit positive integers satisfy the following properties?
The number is divisible by
.
The number formed by reversing the digits of
is divisble by
.
Problem 13
Abdul and Chiang are standing feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chaing measures
. What is the square of the distance (in feet) between Abdul and Bharat?
Problem 14
A number is chosen at random from among the first positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by
?
Problem 15
An even number of circles are nested, starting with a radius of and increasing by
each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius
but outside the circle of radius
An example showing
circles is displayed below. What is the least number of circles needed to make the total shaded area at least
?
See also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2022 Fall AMC 10B Problems |
Followed by 2023 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.