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Difference between revisions of "2025 AMC 8 Problems"

m (→‎Problem 15)
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==Problem 1==
 
==Problem 1==
 
The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire \(4\times4\) grid is covered by the star?
 
The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire \(4\times4\) grid is covered by the star?
 +
 +
 +
<asy>
 +
path x = (0,1)--(1,2)--(2,2)--(1,1)--cycle;
 +
path y = reflect((0,0),(4,4)) * x;
 +
 +
fill(x, gray(0.6));
 +
fill(rotate(90, (2,2)) * x, gray(0.6));
 +
fill(rotate(180, (2,2)) * x, gray(0.6));
 +
fill(rotate(270, (2,2)) * x, gray(0.6));
 +
 +
fill(y, gray(0.8));
 +
fill(rotate(90, (2,2)) * y, gray(0.8));
 +
fill(rotate(180, (2,2)) * y, gray(0.8));
 +
fill(rotate(270, (2,2)) * y, gray(0.8));
 +
 +
draw((1,1)--(3,3));
 +
draw((3,1)--(1,3));
 +
 +
add(grid(4,4));
 +
 +
path w = (1,0)--(2,1)--(3,0);
 +
 +
draw(w);
 +
draw(rotate(90, (2,2)) * w);
 +
draw(rotate(180, (2,2)) * w);
 +
draw(rotate(270, (2,2)) * w);
 +
</asy>
  
 
<math>\textbf{(A)}\ 40 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 60 \qquad \textbf{(D)}\ 75 \qquad \textbf{(E)}\ 80</math>
 
<math>\textbf{(A)}\ 40 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 60 \qquad \textbf{(D)}\ 75 \qquad \textbf{(E)}\ 80</math>
Line 34: Line 62:
  
 
==Problem 5==
 
==Problem 5==
−
[[2025 AMC 8 Problems/Problem 5|Solution]]
 
  
−
==Problem 6==
+
Betty drives a truck to deliver packages in a neighborhood whose street map is shown below.
 +
Betty starts at the factory (labled <math>F</math>) and drives to location <math>A</math>, then <math>B</math>, then <math>C</math>, before returning to <math>F</math>. What is the shortest distance, in blocks, she can drive to complete the route?
 +
 
 +
<asy>
 +
 
 +
unitsize(20);
 +
 
 +
add(grid(8,6));
 +
 
 +
path w = circle((0,0),0.4);
 +
 
 +
fill(w, white);
 +
draw(w);
 +
label("$B$",(0,0));
 +
 
 +
fill(shift((2,4)) * w, white);
 +
draw(shift((2,4)) * w);
 +
label("$C$",(2,4));
 +
 
 +
fill(shift((7,3)) * w, white);
 +
draw(shift((7,3)) * w);
 +
label("$A$",(7,3));
  
−
Sekou writes the numbers <math>15, 16, 17, 18, 19.</math> After he erases one of his numbers, the sum of the remaining four numbers is a multiple of <math>4.</math> Which number did he erase?
+
fill(shift((6,5)) * w, white);
 +
draw(shift((6,5)) * w);
 +
label("$F$",(6,5));
  
−
<math>\textbf{(A)}\ 15\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 19</math>
+
draw((6,-0.2)--(7,-0.2), EndArrow(3));
 +
draw((7,-0.2)--(6,-0.2), EndArrow(3));
 +
draw(shift(6.5, -0.48) * scale(0.03) * texpath("1 block"));
  
−
[[2025 AMC 8 Problems/Problem 6|Solution]]
+
draw((8.2,1)--(8.2,2), EndArrow(3));
 +
draw((8.2,2)--(8.2,1), EndArrow(3));
 +
draw(shift(8.88, 1.5) * scale(0.03) * texpath("1 block"));
  
−
==Problem 7==
+
</asy>
−
On the most recent exam on Prof. Xochi's class,
 
  
 +
<math>\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 22 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 26\qquad \textbf{(E)}\ 28</math>
  
−
5 students earned a score of at least 95%,
+
[[2025 AMC 8 Problems/Problem 5|Solution]]
  
−
13 students earned a score of at least 90%,
+
==Problem 6==
  
−
27 students earned a score of at least 85%,
+
Sekou writes the numbers <math>15, 16, 17, 18, 19.</math> After he erases one of his numbers, the sum of the remaining four numbers is a multiple of <math>4.</math> Which number did he erase?
  
−
50 students earned a score of at least 80%,
+
<math>\textbf{(A)}\ 15\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 19</math>
  
 +
[[2025 AMC 8 Problems/Problem 6|Solution]]
  
 +
==Problem 7==
 +
On the most recent exam on Prof. Xochi's class, 
 +
  <math>5</math> students earned a score of at least <math>95</math>%, 
 +
  <math>13</math> students earned a score of at least <math>90</math>%, 
 +
  <math>27</math> students earned a score of at least <math>85</math>%, 
 +
  <math>50</math> students earned a score of at least <math>80</math>%,
 
How many students earned a score of at least 80% and less than 90%?
 
How many students earned a score of at least 80% and less than 90%?
  
Line 64: Line 125:
  
 
==Problem 8==
 
==Problem 8==
 +
Isaiah cuts open a cardboard cube along some of its edges to form the flat shape shown on the right, which has an area of 18 square centimeters. What is the volume of the cube in cubic centimeters?
 +
 +
<math>\textbf{(A)}~3\sqrt{3}\qquad\textbf{(B)}~6\qquad\textbf{(C)}~9\qquad\textbf{(D)}~6\sqrt{3}\qquad\textbf{(E)}~9\sqrt{3}</math>
 +
 
[[2025 AMC 8 Problems/Problem 8|Solution]]
 
[[2025 AMC 8 Problems/Problem 8|Solution]]
  
 
==Problem 9==
 
==Problem 9==
−
 
  
 
Ningli looks at the 6 pairs of numbers directly across from each other on a clock. She takes the average of each pair of numbers. What is the average of the resulting 6 numbers?
 
Ningli looks at the 6 pairs of numbers directly across from each other on a clock. She takes the average of each pair of numbers. What is the average of the resulting 6 numbers?
Line 76: Line 140:
  
 
==Problem 10==
 
==Problem 10==
 +
 +
 +
In the figure below, <math>ABCD</math> is a rectangle with sides of length <math>AB = 5</math> inches and <math>AD</math> = 3 inches. Rectangle <math>ABCD</math> is rotated <math>90^\circ</math> clockwise around the midpoint of side <math>DC</math> to give a second rectangle. What is the total area, in square inches, covered by the two overlapping rectangles?
 +
 +
<math>\textbf{(A)}\ 21 \qquad \textbf{(B)}\ 22.25 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 23.75 \qquad \textbf{(E)}\ 25</math>
 +
 
[[2025 AMC 8 Problems/Problem 10|Solution]]
 
[[2025 AMC 8 Problems/Problem 10|Solution]]
  
 
==Problem 11==
 
==Problem 11==
 +
 +
A <math>\textit{tetromino}</math> consists of four squares connected along their edges. There are five possible tetromino shapes, <math>I</math>, <math>O</math>, <math>L</math>, <math>T</math>, and <math>S</math>, shown below, which can be rotated or flipped over. Three tetrominoes are used to completely cover a <math>3\times4</math> rectangle. At least one of the tiles is an <math>S</math> tile. What are the other two tiles?
 +
 +
<asy>
 +
 +
unitsize(12);
 +
 +
add(grid(1,4));
 +
label("I", (0.5,-1));
 +
 +
add(shift((5,0)) * grid(2,2));
 +
label("O", (6,-1));
 +
 +
add(shift((11,0)) * grid(1,3));
 +
add(shift((11,0)) * grid(2,1));
 +
label("L", (12,-1));
 +
 +
add(shift((18,0)) * grid(1,1));
 +
add(shift((17,1)) * grid(3,1));
 +
label("T", (18.5,-1));
 +
 +
add(shift((25,1)) * grid(2,1));
 +
add(shift((24,0)) * grid(2,1));
 +
label("S", (25.5,-1));
 +
 +
add(shift((12,-6)) * grid(4,3));
 +
 +
</asy>
 +
 +
<math>\textbf{(A)}I</math> and <math>L\qquad \textbf{(B)} I</math> and <math>T\qquad \textbf{(C)} L</math> and <math>L\qquad \textbf{(D)}L</math> and <math>S\qquad \textbf{(E)}O</math> and <math>T</math>
 +
 
[[2025 AMC 8 Problems/Problem 11|Solution]]
 
[[2025 AMC 8 Problems/Problem 11|Solution]]
  
Line 124: Line 225:
  
 
==Problem 13==
 
==Problem 13==
 +
Each of the even numbers <math>2, 4, 6, \ldots, 50</math> is divided by <math>7</math>. The remainders are recorded. Which histogram displays the number of times each remainder occurs?
 +
 
[[2025 AMC 8 Problems/Problem 13|Solution]]
 
[[2025 AMC 8 Problems/Problem 13|Solution]]
  
 
==Problem 14==
 
==Problem 14==
  
−
A number <math>N</math> is inserted into the list 2, 6, 7, 7, 28. The mean is now twice as great as the median. What is <math>N</math>?
+
A number <math>N</math> is inserted into the list <math>2, 6, 7, 7, 28</math>. The mean is now twice as great as the median. What is <math>N</math>?
  
 
<math>\textbf{(A)}\ 7\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 28\qquad \textbf{(E)}\ 34</math>
 
<math>\textbf{(A)}\ 7\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 28\qquad \textbf{(E)}\ 34</math>
Line 135: Line 238:
  
 
==Problem 15==
 
==Problem 15==
 +
Kei draws a <math>6</math>-by-<math>6</math> grid. He colors <math>13</math> of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let <math>m</math> and <math>M</math> equal the least and greatest possible number of gold-on-gold pairs, respectively. What is the value of <math>m+M</math>?
  
 +
<asy>
 +
import graph;
  
−
Kei draws a 6-by-6 grid. He colors 13 of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let <math>m  and </math>M equal equal the least and greatest possible number of gold-on-gold pairs, respectively. What is the value of <math>m+</math>M?
+
size(100);
 +
 
 +
pen gridPen = black;
 +
 
 +
void drawSquare(pair p) {
 +
    draw(box(p, p + (1,1)), gridPen);
 +
}
 +
 
 +
int[][] grid = {
 +
    {1, 1, 1, 1, 1, 1},
 +
    {1, 1, 1, 1, 1, 1},
 +
    {1, 1, 1, 1, 1, 1},
 +
    {1, 1, 1, 1, 1, 1},
 +
    {1, 1, 1, 1, 1, 1},
 +
    {1, 1, 1, 1, 1, 1},
 +
 
 +
};
 +
 
 +
int rows = grid.length;
 +
int cols = grid[0].length;
 +
 
 +
for (int i = 0; i < rows; ++i) {
 +
    for (int j = 0; j < cols; ++j) {
 +
        if (grid[i][j] == 1) {
 +
            drawSquare((j, rows - i - 1));
 +
        }
 +
    }
 +
}
 +
</asy>
  
 
<math>\textbf{(A)}\ 12\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 20</math>
 
<math>\textbf{(A)}\ 12\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 20</math>
Line 146: Line 280:
  
  
−
Five distinct integers from 1 to 10 are chosen, and five distinct integers from 11 to 20 are chosen. No two numbers differ by exactly 10. What is the sum of the ten chosen numbers?
+
Five distinct integers from <math>1</math> to <math>10</math> are chosen, and five distinct integers from <math>11</math> to <math>20</math> are chosen. No two numbers differ by exactly <math>10</math>. What is the sum of the ten chosen numbers?
  
 
<math>\textbf{(A)}\ 95\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 105\qquad \textbf{(D)}\ 110\qquad \textbf{(E)}\ 115</math>
 
<math>\textbf{(A)}\ 95\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 105\qquad \textbf{(D)}\ 110\qquad \textbf{(E)}\ 115</math>
Line 153: Line 287:
  
 
==Problem 17==
 
==Problem 17==
 +
In the land of Markovia, there are three cities: <math>A</math>, <math>B</math>, and <math>C</math>. There are <math>100</math> people who live in <math>A</math>, <math>120</math> who live in <math>B</math>, and <math>160</math> who live in <math>C</math>. Everyone works in one of the three cities, and a person may work in the same city where they live. In the figure below, an arrow pointing from one city to another is labeled with the fraction of people living in the first city who work in the second city. (For example, <math>\frac{1}{4}</math> of the people who live in <math>A</math> work in <math>B</math>.) How many people work in <math>A</math>? 
 +
 +
<math>\textbf{(A)}\ 55\qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 115\qquad \textbf{(E)}\ 160</math>
 +
 
[[2025 AMC 8 Problems/Problem 17|Solution]]
 
[[2025 AMC 8 Problems/Problem 17|Solution]]
  
Line 159: Line 297:
  
 
The circle shown below on the left has a radius of 1 unit. The region between the circle and the inscribed square is shaded. In the circle shown on the right, one quarter of the region between the circle and the inscribed square is shaded. The shaded regions in the two circles have the same area. What is the radius <math>R</math>, in units, of the circle on the right?
 
The circle shown below on the left has a radius of 1 unit. The region between the circle and the inscribed square is shaded. In the circle shown on the right, one quarter of the region between the circle and the inscribed square is shaded. The shaded regions in the two circles have the same area. What is the radius <math>R</math>, in units, of the circle on the right?
 +
 +
<asy>
 +
 +
unitsize(40);
 +
 +
real a = 0.707;
 +
 +
fill(circle((a,a), 1), grey);
 +
fill((0,0)--(0,1.414)--(1.414,1.414)--(1.414,0)--cycle, white);
 +
draw((0,0)--(0,1.414)--(1.414,1.414)--(1.414,0)--cycle);
 +
draw(circle((a,a), 1));
 +
 +
draw((0.707,0.707)--(1.414,1.414));
 +
dot((0.707,0.707));
 +
label("$1$", (1,1), SE);
 +
 +
 +
 +
fill(circle((4+a, a), 2*a), grey);
 +
fill(shift((4+a,a)) * ((-2,-2)--(1,-2)--(1,2)--(-2,2)--cycle), white);
 +
draw(shift((4+a,a)) * ((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle));
 +
draw(circle((4+a, a), 2*a));
 +
 +
draw((4+a,a)--(5+a,1+a));
 +
dot((4+a,a));
 +
label("$R$", (a+4.5,a+0.5), SE);
 +
 +
</asy>
  
 
<math>\textbf{(A)}\ \sqrt2\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 2\sqrt2\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 4\sqrt2</math>
 
<math>\textbf{(A)}\ \sqrt2\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 2\sqrt2\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 4\sqrt2</math>
Line 165: Line 331:
  
 
==Problem 19==
 
==Problem 19==
 +
 +
 +
Two towns, <math>A</math> and <math>B</math>, are connected by a straight road, <math>15</math> miles long. Traveling from town <math>A</math> to town <math>B</math>, the speed limit changes every <math>5</math> miles: from <math>25</math> to <math>40</math> to <math>20</math> miles per hour (mph). Two cars, one at town <math>A</math> and one at town <math>B</math>, start moving toward each other at the same time. They drive at exactly the speed limit in each portion of the road. How far from town <math>A</math>, in miles, will the two cars meet?
 +
 +
<math>\textbf{(A)}\ 7.75\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 8.25\qquad \textbf{(D)}\ 8.5\qquad \textbf{(E)}\ 8.75</math>
 +
 
[[2025 AMC 8 Problems/Problem 19|Solution]]
 
[[2025 AMC 8 Problems/Problem 19|Solution]]
  
Line 185: Line 357:
 
levels. (For example, grades 1 and 2 will not be in pods directly connected by a walkway.) What is
 
levels. (For example, grades 1 and 2 will not be in pods directly connected by a walkway.) What is
 
the sum of the grade levels assigned to pods <math>C, E</math>, and <math>F</math>?
 
the sum of the grade levels assigned to pods <math>C, E</math>, and <math>F</math>?
−
 
  
 
<math>\textbf{(A)}~12\qquad\textbf{(B)}~13\qquad\textbf{(C)}~14\qquad\textbf{(D)}~15\qquad\textbf{(E)}~16</math>
 
<math>\textbf{(A)}~12\qquad\textbf{(B)}~13\qquad\textbf{(C)}~14\qquad\textbf{(D)}~15\qquad\textbf{(E)}~16</math>
Line 214: Line 385:
  
 
==Problem 24==
 
==Problem 24==
−
In trapezoid <math>ABCD</math>, angles <math>B</math> and <math>C</math> measure <math>60^\circ</math> and <math>AB = DC</math>. The side lengths are all positive integers, and the perimeter of <math>ABCD</math> is 30 units. How many non-congruent trapezoids satisfy all of these conditions?
+
In trapezoid <math>ABCD</math>, angles <math>B</math> and <math>C</math> measure <math>60^\circ</math> and <math>AB = DC</math>. The side lengths are all positive integers, and the perimeter of <math>ABCD</math> is <math>30</math> units. How many non-congruent trapezoids satisfy all of these conditions?
  
 
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4</math>
 
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4</math>
Line 222: Line 393:
 
==Problem 25==
 
==Problem 25==
 
Makayla finds all the possible ways to draw a path in a <math>5 \times 5</math> diamond-shaped grid. Each path starts at the bottom of the grid and ends at the top, always moving one unit northeast or northwest. She computes the area of the region between each path and the right side of the grid. Two examples are shown in the figures below. What is the sum of the areas determined by all possible paths?
 
Makayla finds all the possible ways to draw a path in a <math>5 \times 5</math> diamond-shaped grid. Each path starts at the bottom of the grid and ends at the top, always moving one unit northeast or northwest. She computes the area of the region between each path and the right side of the grid. Two examples are shown in the figures below. What is the sum of the areas determined by all possible paths?
 +
 +
<asy>
 +
unitsize(9);
 +
 +
real a = 0.7071;
 +
 +
path w = (0,0)--(2a, 2a)--(-a,5a)--(a,7a)--(-a,9a)--(0,10a);
 +
 +
fill(w--(5a,5a)--cycle, gray(0.8));
 +
draw(w, linewidth(1.5));
 +
 +
path x = (10,0)--(10-a,a)--(10+2a,4a)--(10-2a,8a)--(10,10a);
 +
 +
fill(x--(10+5a,5a)--cycle, gray(0.8));
 +
draw(x, linewidth(1.5));
 +
 +
add(rotate(45, (0,0)) * grid(5,5));
 +
add(rotate(45, (10,0)) * (shift((10,0)) * grid(5,5)));
 +
 +
dot((0,0));
 +
dot((0,7.07106));
 +
dot((10,0));
 +
dot((10,7.07106));
 +
 +
label("area = 11", (0,-1), S);
 +
label("area = 13", (10,-1), S);
 +
</asy>
 +
  
 
<math>\textbf{(A)}\ 2520 \qquad \textbf{(B)}\ 3150 \qquad \textbf{(C)}\ 3840 \qquad \textbf{(D)}\ 4730 \qquad \textbf{(E)}\ 5050</math>
 
<math>\textbf{(A)}\ 2520 \qquad \textbf{(B)}\ 3150 \qquad \textbf{(C)}\ 3840 \qquad \textbf{(D)}\ 4730 \qquad \textbf{(E)}\ 5050</math>

Latest revision as of 23:41, 30 January 2025


2025 AMC 8 (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. 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.
  2. You will receive 1 point for each correct answer. There is no penalty for wrong answers.
  3. No aids are permitted other than plain scratch paper, writing utensils, ruler, and erasers. In particular, graph paper, compass, protractor, calculators, computers, smartwatches, and smartphones are not permitted. Rules
  4. Figures are not necessarily drawn to scale.
  5. You will have 40 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

The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire \(4\times4\) grid is covered by the star?


[asy] path x = (0,1)--(1,2)--(2,2)--(1,1)--cycle; path y = reflect((0,0),(4,4)) * x; fill(x, gray(0.6)); fill(rotate(90, (2,2)) * x, gray(0.6)); fill(rotate(180, (2,2)) * x, gray(0.6)); fill(rotate(270, (2,2)) * x, gray(0.6)); fill(y, gray(0.8)); fill(rotate(90, (2,2)) * y, gray(0.8)); fill(rotate(180, (2,2)) * y, gray(0.8)); fill(rotate(270, (2,2)) * y, gray(0.8)); draw((1,1)--(3,3)); draw((3,1)--(1,3)); add(grid(4,4)); path w = (1,0)--(2,1)--(3,0); draw(w); draw(rotate(90, (2,2)) * w); draw(rotate(180, (2,2)) * w); draw(rotate(270, (2,2)) * w); [/asy]

$\textbf{(A)}\ 40 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 60 \qquad \textbf{(D)}\ 75 \qquad \textbf{(E)}\ 80$

Solution

Problem 2

The table below shows the ancient Egyptian hieroglyphs that were used to represent different numbers.

For example, the number $32$ was represented by the hieroglyphs $\cap \cap \cap ||$. What number is represented by the following combination of hieroglyphs?

$\textbf{(A)}\ 1,423 \qquad \textbf{(B)}\ 10,423 \qquad \textbf{(C)}\ 14,023 \qquad \textbf{(D)}\ 14,203 \qquad \textbf{(E)}\ 14,230$

Solution

Problem 3

Buffalo Shuffle-o is a card game in which all the cards are distributed evenly among all players at the start of the game. When Annika and $3$ of her friends play Buffalo Shuffle-o, each player is dealt $15$ cards. Suppose $2$ more friends join the next game. How many cards will be dealt to each player?

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

Solution

Problem 4

Lucius is counting backward by $7$s. His first three numbers are $100$, $93$, and $86$. What is his $10$th number?

$\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 42 \qquad \textbf{(D)}\ 44 \qquad \textbf{(E)}\ 47$

Solution

Problem 5

Betty drives a truck to deliver packages in a neighborhood whose street map is shown below. Betty starts at the factory (labled $F$) and drives to location $A$, then $B$, then $C$, before returning to $F$. What is the shortest distance, in blocks, she can drive to complete the route?

[asy] unitsize(20); add(grid(8,6)); path w = circle((0,0),0.4); fill(w, white); draw(w); label("$B$",(0,0)); fill(shift((2,4)) * w, white); draw(shift((2,4)) * w); label("$C$",(2,4)); fill(shift((7,3)) * w, white); draw(shift((7,3)) * w); label("$A$",(7,3)); fill(shift((6,5)) * w, white); draw(shift((6,5)) * w); label("$F$",(6,5)); draw((6,-0.2)--(7,-0.2), EndArrow(3)); draw((7,-0.2)--(6,-0.2), EndArrow(3)); draw(shift(6.5, -0.48) * scale(0.03) * texpath("1 block")); draw((8.2,1)--(8.2,2), EndArrow(3)); draw((8.2,2)--(8.2,1), EndArrow(3)); draw(shift(8.88, 1.5) * scale(0.03) * texpath("1 block")); [/asy]

$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 22 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 26\qquad \textbf{(E)}\ 28$

Solution

Problem 6

Sekou writes the numbers $15, 16, 17, 18, 19.$ After he erases one of his numbers, the sum of the remaining four numbers is a multiple of $4.$ Which number did he erase?

$\textbf{(A)}\ 15\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 19$

Solution

Problem 7

On the most recent exam on Prof. Xochi's class, 

 $5$ students earned a score of at least $95$%,  
 $13$ students earned a score of at least $90$%,  
 $27$ students earned a score of at least $85$%,  
 $50$ students earned a score of at least $80$%,

How many students earned a score of at least 80% and less than 90%?

$\textbf{(A)}\ 8\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 37\qquad \textbf{(E)}\ 45$

Solution

Problem 8

Isaiah cuts open a cardboard cube along some of its edges to form the flat shape shown on the right, which has an area of 18 square centimeters. What is the volume of the cube in cubic centimeters?

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

Solution

Problem 9

Ningli looks at the 6 pairs of numbers directly across from each other on a clock. She takes the average of each pair of numbers. What is the average of the resulting 6 numbers?

$\textbf{(A)}\ 5\qquad \textbf{(B)}\ 6.5\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9.5 \qquad \textbf{(E)}\ 12$

Solution

Problem 10

In the figure below, $ABCD$ is a rectangle with sides of length $AB = 5$ inches and $AD$ = 3 inches. Rectangle $ABCD$ is rotated $90^\circ$ clockwise around the midpoint of side $DC$ to give a second rectangle. What is the total area, in square inches, covered by the two overlapping rectangles?

$\textbf{(A)}\ 21 \qquad \textbf{(B)}\ 22.25 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 23.75 \qquad \textbf{(E)}\ 25$

Solution

Problem 11

A $\textit{tetromino}$ consists of four squares connected along their edges. There are five possible tetromino shapes, $I$, $O$, $L$, $T$, and $S$, shown below, which can be rotated or flipped over. Three tetrominoes are used to completely cover a $3\times4$ rectangle. At least one of the tiles is an $S$ tile. What are the other two tiles?

[asy] unitsize(12); add(grid(1,4)); label("I", (0.5,-1)); add(shift((5,0)) * grid(2,2)); label("O", (6,-1)); add(shift((11,0)) * grid(1,3)); add(shift((11,0)) * grid(2,1)); label("L", (12,-1)); add(shift((18,0)) * grid(1,1)); add(shift((17,1)) * grid(3,1)); label("T", (18.5,-1)); add(shift((25,1)) * grid(2,1)); add(shift((24,0)) * grid(2,1)); label("S", (25.5,-1)); add(shift((12,-6)) * grid(4,3)); [/asy]

$\textbf{(A)}I$ and $L\qquad \textbf{(B)} I$ and $T\qquad \textbf{(C)} L$ and $L\qquad \textbf{(D)}L$ and $S\qquad \textbf{(E)}O$ and $T$

Solution

Problem 12

The region shown below consists of 24 squares, each with side length 1 centimeter. What is the area, in square centimeters, of the largest circle that can fit inside the region, possibly touching the boundaries?

[asy] import graph; size(100); pen gridPen = black; void drawSquare(pair p) { draw(box(p, p + (1,1)), gridPen); } int[][] grid = { {0, 0, 0, 0, 0, 0}, {0, 0, 1, 1, 0, 0}, {0, 1, 1, 1, 1, 0}, {1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}, {0, 1, 1, 1, 1, 0}, {0, 0, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 0} }; int rows = grid.length; int cols = grid[0].length; for (int i = 0; i < rows; ++i) { for (int j = 0; j < cols; ++j) { if (grid[i][j] == 1) { drawSquare((j, rows - i - 1)); } } } [/asy]

$\textbf{(A)}\ 3\pi\qquad \textbf{(B)}\ 4\pi\qquad \textbf{(C)}\ 5\pi\qquad \textbf{(D)}\ 6\pi\qquad \textbf{(E)}\ 8\pi$

Solution

Problem 13

Each of the even numbers $2, 4, 6, \ldots, 50$ is divided by $7$. The remainders are recorded. Which histogram displays the number of times each remainder occurs?

Solution

Problem 14

A number $N$ is inserted into the list $2, 6, 7, 7, 28$. The mean is now twice as great as the median. What is $N$?

$\textbf{(A)}\ 7\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 28\qquad \textbf{(E)}\ 34$

Solution

Problem 15

Kei draws a $6$-by-$6$ grid. He colors $13$ of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let $m$ and $M$ equal the least and greatest possible number of gold-on-gold pairs, respectively. What is the value of $m+M$?

[asy] import graph; size(100); pen gridPen = black; void drawSquare(pair p) { draw(box(p, p + (1,1)), gridPen); } int[][] grid = { {1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}, }; int rows = grid.length; int cols = grid[0].length; for (int i = 0; i < rows; ++i) { for (int j = 0; j < cols; ++j) { if (grid[i][j] == 1) { drawSquare((j, rows - i - 1)); } } } [/asy]

$\textbf{(A)}\ 12\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 20$

Solution

Problem 16

Five distinct integers from $1$ to $10$ are chosen, and five distinct integers from $11$ to $20$ are chosen. No two numbers differ by exactly $10$. What is the sum of the ten chosen numbers?

$\textbf{(A)}\ 95\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 105\qquad \textbf{(D)}\ 110\qquad \textbf{(E)}\ 115$

Solution

Problem 17

In the land of Markovia, there are three cities: $A$, $B$, and $C$. There are $100$ people who live in $A$, $120$ who live in $B$, and $160$ who live in $C$. Everyone works in one of the three cities, and a person may work in the same city where they live. In the figure below, an arrow pointing from one city to another is labeled with the fraction of people living in the first city who work in the second city. (For example, $\frac{1}{4}$ of the people who live in $A$ work in $B$.) How many people work in $A$?

$\textbf{(A)}\ 55\qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 115\qquad \textbf{(E)}\ 160$

Solution

Problem 18

The circle shown below on the left has a radius of 1 unit. The region between the circle and the inscribed square is shaded. In the circle shown on the right, one quarter of the region between the circle and the inscribed square is shaded. The shaded regions in the two circles have the same area. What is the radius $R$, in units, of the circle on the right?

[asy] unitsize(40); real a = 0.707; fill(circle((a,a), 1), grey); fill((0,0)--(0,1.414)--(1.414,1.414)--(1.414,0)--cycle, white); draw((0,0)--(0,1.414)--(1.414,1.414)--(1.414,0)--cycle); draw(circle((a,a), 1)); draw((0.707,0.707)--(1.414,1.414)); dot((0.707,0.707)); label("$1$", (1,1), SE); fill(circle((4+a, a), 2*a), grey); fill(shift((4+a,a)) * ((-2,-2)--(1,-2)--(1,2)--(-2,2)--cycle), white); draw(shift((4+a,a)) * ((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle)); draw(circle((4+a, a), 2*a)); draw((4+a,a)--(5+a,1+a)); dot((4+a,a)); label("$R$", (a+4.5,a+0.5), SE); [/asy]

$\textbf{(A)}\ \sqrt2\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 2\sqrt2\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 4\sqrt2$

Solution

Problem 19

Two towns, $A$ and $B$, are connected by a straight road, $15$ miles long. Traveling from town $A$ to town $B$, the speed limit changes every $5$ miles: from $25$ to $40$ to $20$ miles per hour (mph). Two cars, one at town $A$ and one at town $B$, start moving toward each other at the same time. They drive at exactly the speed limit in each portion of the road. How far from town $A$, in miles, will the two cars meet?

$\textbf{(A)}\ 7.75\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 8.25\qquad \textbf{(D)}\ 8.5\qquad \textbf{(E)}\ 8.75$

Solution

Problem 20

Sarika, Dev, and Rajiv are sharing a large block of cheese. They take turns cutting off half of what remains and eating it: first Sarika eats half of the cheese, then Dev eats half of the remaining half, then Rajiv eats half of what remains, then back to Sarika, and so on. They stop when the cheese is too small to see. About what fraction of the original block of cheese does Sarika eat in total?

$\textbf{(A)}\ \frac{4}{7}\qquad \textbf{(B)}\ \frac{3}{5}\qquad \textbf{(C)}\ \frac{2}{3}\qquad \textbf{(D)}\ \frac{3}{4}\qquad \textbf{(E)}\ \frac{7}{8}$

Solution

Problem 21

The Konigsberg School has assigned grades 1 through 7 to pods $A$ through $G$, one grade per pod. Some of the pods are connected by walkways, as shown in the figure below. The school noticed that each pair of connected pods has been assigned grades differing by 2 or more grade levels. (For example, grades 1 and 2 will not be in pods directly connected by a walkway.) What is the sum of the grade levels assigned to pods $C, E$, and $F$?

$\textbf{(A)}~12\qquad\textbf{(B)}~13\qquad\textbf{(C)}~14\qquad\textbf{(D)}~15\qquad\textbf{(E)}~16$

Solution

Problem 22

A classroom has a row of 35 coat hooks. Paulina likes coats to be equally spaced, so that there is the same number of empty hooks before the first coat, after the last coat, and between every coat and the next one. Suppose there is at least 1 coat and at least 1 empty hook. How many different numbers of coats can satisfy Paulina's pattern?

$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 9$

Solution

Problem 23

How many four-digit numbers have all three of the following properties?

(I) The tens and ones digit are both 9.

(II) The number is 1 less than a perfect square.

(III) The number is the product of exactly two prime numbers.

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Solution

Problem 24

In trapezoid $ABCD$, angles $B$ and $C$ measure $60^\circ$ and $AB = DC$. The side lengths are all positive integers, and the perimeter of $ABCD$ is $30$ units. How many non-congruent trapezoids satisfy all of these conditions?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Solution

Problem 25

Makayla finds all the possible ways to draw a path in a $5 \times 5$ diamond-shaped grid. Each path starts at the bottom of the grid and ends at the top, always moving one unit northeast or northwest. She computes the area of the region between each path and the right side of the grid. Two examples are shown in the figures below. What is the sum of the areas determined by all possible paths?

[asy] unitsize(9); real a = 0.7071; path w = (0,0)--(2a, 2a)--(-a,5a)--(a,7a)--(-a,9a)--(0,10a); fill(w--(5a,5a)--cycle, gray(0.8)); draw(w, linewidth(1.5)); path x = (10,0)--(10-a,a)--(10+2a,4a)--(10-2a,8a)--(10,10a); fill(x--(10+5a,5a)--cycle, gray(0.8)); draw(x, linewidth(1.5)); add(rotate(45, (0,0)) * grid(5,5)); add(rotate(45, (10,0)) * (shift((10,0)) * grid(5,5))); dot((0,0)); dot((0,7.07106)); dot((10,0)); dot((10,7.07106)); label("area = 11", (0,-1), S); label("area = 13", (10,-1), S); [/asy]


$\textbf{(A)}\ 2520 \qquad \textbf{(B)}\ 3150 \qquad \textbf{(C)}\ 3840 \qquad \textbf{(D)}\ 4730 \qquad \textbf{(E)}\ 5050$

Solution

See Also

2025 AMC 8 (Problems • Answer Key • Resources)
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