Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
ONLINE AMC 8 PREP WITH AOPS
Top scorers around the country use AoPS. Join training courses for beginners and advanced students.
VIEW CATALOG

Difference between revisions of "2023 AMC 8 Problems"

(→‎Problem 12)
(Undo revision 228745 by Ben.iriarte (talk))
(Tag: Undo)
 
(150 intermediate revisions by 42 users not shown)
Line 1: Line 1:
 
{{AMC8 Problems|year=2023|}}
 
{{AMC8 Problems|year=2023|}}
−
 
−
TODO: transcribe from [https://cdn.artofproblemsolving.com/attachments/9/f/7d6170e017d291e9eb965bee17eacf4a670a62.pdf]
 
−
 
 
==Problem 1==
 
==Problem 1==
  
Line 13: Line 10:
 
==Problem 2==
 
==Problem 2==
  
−
When a square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?
+
A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?
 +
<asy>
 +
//Restored original diagram. Alter it if you would like, but it was made by TheMathGuyd,
 +
// Diagram by TheMathGuyd. I even put the lined texture :)
 +
// Thank you Kante314 for inspiring thicker arrows. They do look much better
 +
size(0,3cm);
 +
path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle;
 +
path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle;
 +
path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle;
 +
path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle;
 +
path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle;
 +
path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle;
 +
path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle;
 +
filldraw(sq,mediumgrey,black);
 +
draw((0.75,0)--(1.25,0),currentpen+1,Arrow(size=6));
 +
//folding
 +
path sqside = (-0.5,-0.5)--(0.5,-0.5);
 +
path rhside = (-0.125,-0.125)--(0.5,-0.5);
 +
transform fld = shift((1.75,0))*scale(0.5);
 +
draw(fld*sq,black);
 +
int i;
 +
for(i=0; i<10; i=i+1)
 +
{
 +
  draw(shift(0,0.05*i)*fld*sqside,deepblue);
 +
}
 +
path rhedge = (-0.125,-0.125)--(-0.125,0.8)--(-0.2,0.85)--cycle;
 +
filldraw(fld*rhedge,grey);
 +
path sqedge = (-0.5,-0.5)--(-0.5,0.4475)--(-0.575,0.45)--cycle;
 +
filldraw(fld*sqedge,grey);
 +
filldraw(fld*rh,white,black);
 +
int i;
 +
for(i=0; i<10; i=i+1)
 +
{
 +
  draw(shift(0,0.05*i)*fld*rhside,deepblue);
 +
}
 +
draw((2.25,0)--(2.75,0),currentpen+1,Arrow(size=6));
 +
//cutting
 +
transform cut = shift((3.25,0))*scale(0.5);
 +
draw(shift((-0.01,+0.01))*cut*sq);
 +
draw(cut*sq);
 +
filldraw(shift((0.01,-0.01))*cut*sq,white,black);
 +
int j;
 +
for(j=0; j<10; j=j+1)
 +
{
 +
draw(shift(0,0.05*j)*cut*sqside,deepblue);
 +
}
 +
draw(shift((0.01,-0.01))*cut*(0,-0.5)--shift((0.01,-0.01))*cut*(0.5,0),dashed);
 +
//Answers Below, but already Separated
 +
//filldraw(sqA,grey,black);
 +
//filldraw(sqB,grey,black);
 +
//filldraw(sq,grey,black);
 +
//filldraw(sqC,white,black);
 +
//filldraw(sq,grey,black);
 +
//filldraw(trD,white,black);
 +
//filldraw(sq,grey,black);
 +
//filldraw(sqE,white,black);
 +
</asy>
  
−
[[Image:2023 AMC 8-2.png|thumb|center|800px]]
+
<asy>
 +
// Diagram by TheMathGuyd.
 +
size(0,7.5cm);
 +
path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle;
 +
path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle;
 +
path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle;
 +
path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle;
 +
path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle;
 +
path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle;
 +
path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle;
  
−
<math>\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X</math>
+
//ANSWERS
 +
real sh = 1.5;
 +
label("$\textbf{(A)}$",(-0.5,0.5),SW);
 +
label("$\textbf{(B)}$",shift((sh,0))*(-0.5,0.5),SW);
 +
label("$\textbf{(C)}$",shift((2sh,0))*(-0.5,0.5),SW);
 +
label("$\textbf{(D)}$",shift((0,-sh))*(-0.5,0.5),SW);
 +
label("$\textbf{(E)}$",shift((sh,-sh))*(-0.5,0.5),SW);
 +
filldraw(sqA,mediumgrey,black);
 +
filldraw(shift((sh,0))*sqB,mediumgrey,black);
 +
filldraw(shift((2*sh,0))*sq,mediumgrey,black);
 +
filldraw(shift((2*sh,0))*sqC,white,black);
 +
filldraw(shift((0,-sh))*sq,mediumgrey,black);
 +
filldraw(shift((0,-sh))*trD,white,black);
 +
filldraw(shift((sh,-sh))*sq,mediumgrey,black);
 +
filldraw(shift((sh,-sh))*sqE,white,black);
 +
</asy>
  
 
[[2023 AMC 8 Problems/Problem 2|Solution]]
 
[[2023 AMC 8 Problems/Problem 2|Solution]]
Line 25: Line 102:
 
<i>Wind chill</i> is a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be found using this calculation
 
<i>Wind chill</i> is a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be found using this calculation
 
<cmath>(\text{wind chill}) = (\text{air temperature}) - 0.7 \times (\text{wind speed}),</cmath>
 
<cmath>(\text{wind chill}) = (\text{air temperature}) - 0.7 \times (\text{wind speed}),</cmath>
−
where temperature is measured in degrees Fahrenheit <math>(^{\circ}\text{F})</math> and and the wind speed is measured in miles per hour (mph). Suppose the air temperature is <math>36^{\circ}\text{F} </math> and the wind speed is <math>18</math> mph. Which of the following is closest to the approximate wind chill?
+
where temperature is measured in degrees Fahrenheit <math>(^{\circ}\text{F})</math> and the wind speed is measured in miles per hour (mph). Suppose the air temperature is <math>36^{\circ}\text{F} </math> and the wind speed is <math>18</math> mph. Which of the following is closest to the approximate wind chill?
  
−
<math>\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 28 \qquad \textbf{(B)}\ 32 \qquad \textbf{(E)}\ 35</math>
+
<math>\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 35</math>
  
 
[[2023 AMC 8 Problems/Problem 3|Solution]]
 
[[2023 AMC 8 Problems/Problem 3|Solution]]
  
 
==Problem 4==
 
==Problem 4==
−
The numbers from 1 to 49 are arranged in a spiral pattern on a square grid, beginning at the center. The
+
The numbers from <math>1</math> to <math>49</math> are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number <math>7.</math> How many of these four numbers are prime?
−
first few numbers have been entered into the grid below. Consider the four numbers that will appear
+
<asy>
−
in the shaded squares, on the same diagonal as the number 7. How many of these four numbers are
+
/* Made by MRENTHUSIASM */
−
prime?
+
size(175);
 +
 
 +
void ds(pair p) {
 +
filldraw((0.5,0.5)+p--(-0.5,0.5)+p--(-0.5,-0.5)+p--(0.5,-0.5)+p--cycle,mediumgrey);
 +
}
 +
 
 +
ds((0.5,4.5));
 +
ds((1.5,3.5));
 +
ds((3.5,1.5));
 +
ds((4.5,0.5));
 +
 
 +
add(grid(7,7,grey+linewidth(1.25)));
 +
 
 +
int adj = 1;
 +
int curUp = 2;
 +
int curLeft = 4;
 +
int curDown = 6;
 +
 
 +
label("$1$",(3.5,3.5));
 +
 
 +
for (int len = 3; len<=3; len+=2)
 +
{
 +
for (int i=1; i<=len-1; ++i)
 +
  {
 +
label("$"+string(curUp)+"$",(3.5+adj,3.5-adj+i));
 +
    label("$"+string(curLeft)+"$",(3.5+adj-i,3.5+adj));
 +
    label("$"+string(curDown)+"$",(3.5-adj,3.5+adj-i));
 +
    ++curDown;
 +
    ++curLeft;
 +
    ++curUp;
 +
}
 +
++adj;
 +
    curUp = len^2 + 1;
 +
    curLeft = len^2 + len + 2;
 +
    curDown = len^2 + 2*len + 3;
 +
}
  
 +
draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(2));
 +
</asy>
 
<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 50: Line 164:
  
 
==Problem 6==
 
==Problem 6==
−
The digits 2, 0, 2, and 3 are placed in the expression below, one digit per box. What is the maximum
+
The digits <math>2, 0, 2,</math> and <math>3</math> are placed in the expression below, one digit per box. What is the maximum possible value of the expression?
−
possible value of the expression?
 
  
−
[[File:2023 AMC 8-6.png|200px|thumb|center]]
+
<asy>
 +
// Diagram by TheMathGuyd. I can compress this later
 +
size(5cm);
 +
real w=2.2;
 +
pair O,I,J;
 +
O=(0,0);I=(1,0);J=(0,1);
 +
path bsqb = O--I;
 +
path bsqr = I--I+J;
 +
path bsqt = I+J--J;
 +
path bsql = J--O;
 +
path lsqb = shift((1.2,0.75))*scale(0.5)*bsqb;
 +
path lsqr = shift((1.2,0.75))*scale(0.5)*bsqr;
 +
path lsqt = shift((1.2,0.75))*scale(0.5)*bsqt;
 +
path lsql = shift((1.2,0.75))*scale(0.5)*bsql;
 +
draw(bsqb,dashed);
 +
draw(bsqr,dashed);
 +
draw(bsqt,dashed);
 +
draw(bsql,dashed);
 +
draw(lsqb,dashed);
 +
draw(lsqr,dashed);
 +
draw(lsqt,dashed);
 +
draw(lsql,dashed);
 +
label(scale(3)*"$\times$",(w,1/3));
 +
draw(shift(1.3w,0)*bsqb,dashed);
 +
draw(shift(1.3w,0)*bsqr,dashed);
 +
draw(shift(1.3w,0)*bsqt,dashed);
 +
draw(shift(1.3w,0)*bsql,dashed);
 +
draw(shift(1.3w,0)*lsqb,dashed);
 +
draw(shift(1.3w,0)*lsqr,dashed);
 +
draw(shift(1.3w,0)*lsqt,dashed);
 +
draw(shift(1.3w,0)*lsql,dashed);
 +
</asy>
  
 
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 18</math>
 
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 18</math>
Line 61: Line 205:
 
==Problem 7==
 
==Problem 7==
  
−
A rectangle, with sides parallel to the <math>x</math>-axis and <math>y</math>-axis, has opposite vertices located at <math>(15, 3)</math> and <math>(16, 5)</math>. A line drawn through points <math>A(0, 0)</math> and <math>B(3, 1)</math>. Another line is drawn through points <math>C(0, 10)</math> and <math>D(2, 9)</math>. How many points on the rectangle lie on at least one of the two lines?
+
A rectangle, with sides parallel to the <math>x</math>-axis and <math>y</math>-axis, has opposite vertices located at <math>(15, 3)</math> and <math>(16, 5)</math>. A line is drawn through points <math>A(0, 0)</math> and <math>B(3, 1)</math>. Another line is drawn through points <math>C(0, 10)</math> and <math>D(2, 9)</math>. How many points on the rectangle lie on at least one of the two lines?
 +
<asy>
 +
usepackage("mathptmx");
 +
size(9cm);
 +
draw((0,-.5)--(0,11),EndArrow(size=.15cm));
 +
draw((1,0)--(1,11),mediumgray);
 +
draw((2,0)--(2,11),mediumgray);
 +
draw((3,0)--(3,11),mediumgray);
 +
draw((4,0)--(4,11),mediumgray);
 +
draw((5,0)--(5,11),mediumgray);
 +
draw((6,0)--(6,11),mediumgray);
 +
draw((7,0)--(7,11),mediumgray);
 +
draw((8,0)--(8,11),mediumgray);
 +
draw((9,0)--(9,11),mediumgray);
 +
draw((10,0)--(10,11),mediumgray);
 +
draw((11,0)--(11,11),mediumgray);
 +
draw((12,0)--(12,11),mediumgray);
 +
draw((13,0)--(13,11),mediumgray);
 +
draw((14,0)--(14,11),mediumgray);
 +
draw((15,0)--(15,11),mediumgray);
 +
draw((16,0)--(16,11),mediumgray);
  
 +
draw((-.5,0)--(17,0),EndArrow(size=.15cm));
 +
draw((0,1)--(17,1),mediumgray);
 +
draw((0,2)--(17,2),mediumgray);
 +
draw((0,3)--(17,3),mediumgray);
 +
draw((0,4)--(17,4),mediumgray);
 +
draw((0,5)--(17,5),mediumgray);
 +
draw((0,6)--(17,6),mediumgray);
 +
draw((0,7)--(17,7),mediumgray);
 +
draw((0,8)--(17,8),mediumgray);
 +
draw((0,9)--(17,9),mediumgray);
 +
draw((0,10)--(17,10),mediumgray);
 +
 +
draw((-.13,1)--(.13,1));
 +
draw((-.13,2)--(.13,2));
 +
draw((-.13,3)--(.13,3));
 +
draw((-.13,4)--(.13,4));
 +
draw((-.13,5)--(.13,5));
 +
draw((-.13,6)--(.13,6));
 +
draw((-.13,7)--(.13,7));
 +
draw((-.13,8)--(.13,8));
 +
draw((-.13,9)--(.13,9));
 +
draw((-.13,10)--(.13,10));
 +
 +
draw((1,-.13)--(1,.13));
 +
draw((2,-.13)--(2,.13));
 +
draw((3,-.13)--(3,.13));
 +
draw((4,-.13)--(4,.13));
 +
draw((5,-.13)--(5,.13));
 +
draw((6,-.13)--(6,.13));
 +
draw((7,-.13)--(7,.13));
 +
draw((8,-.13)--(8,.13));
 +
draw((9,-.13)--(9,.13));
 +
draw((10,-.13)--(10,.13));
 +
draw((11,-.13)--(11,.13));
 +
draw((12,-.13)--(12,.13));
 +
draw((13,-.13)--(13,.13));
 +
draw((14,-.13)--(14,.13));
 +
draw((15,-.13)--(15,.13));
 +
draw((16,-.13)--(16,.13));
 +
 +
label(scale(.7)*"$1$", (1,-.13), S);
 +
label(scale(.7)*"$2$", (2,-.13), S);
 +
label(scale(.7)*"$3$", (3,-.13), S);
 +
label(scale(.7)*"$4$", (4,-.13), S);
 +
label(scale(.7)*"$5$", (5,-.13), S);
 +
label(scale(.7)*"$6$", (6,-.13), S);
 +
label(scale(.7)*"$7$", (7,-.13), S);
 +
label(scale(.7)*"$8$", (8,-.13), S);
 +
label(scale(.7)*"$9$", (9,-.13), S);
 +
label(scale(.7)*"$10$", (10,-.13), S);
 +
label(scale(.7)*"$11$", (11,-.13), S);
 +
label(scale(.7)*"$12$", (12,-.13), S);
 +
label(scale(.7)*"$13$", (13,-.13), S);
 +
label(scale(.7)*"$14$", (14,-.13), S);
 +
label(scale(.7)*"$15$", (15,-.13), S);
 +
label(scale(.7)*"$16$", (16,-.13), S);
 +
 +
label(scale(.7)*"$1$", (-.13,1), W);
 +
label(scale(.7)*"$2$", (-.13,2), W);
 +
label(scale(.7)*"$3$", (-.13,3), W);
 +
label(scale(.7)*"$4$", (-.13,4), W);
 +
label(scale(.7)*"$5$", (-.13,5), W);
 +
label(scale(.7)*"$6$", (-.13,6), W);
 +
label(scale(.7)*"$7$", (-.13,7), W);
 +
label(scale(.7)*"$8$", (-.13,8), W);
 +
label(scale(.7)*"$9$", (-.13,9), W);
 +
label(scale(.7)*"$10$", (-.13,10), W);
 +
 +
 +
dot((0,0),linewidth(4));
 +
label(scale(.75)*"$A$", (0,0), NE);
 +
dot((3,1),linewidth(4));
 +
label(scale(.75)*"$B$", (3,1), NE);
 +
 +
dot((0,10),linewidth(4));
 +
label(scale(.75)*"$C$", (0,10), NE);
 +
dot((2,9),linewidth(4));
 +
label(scale(.75)*"$D$", (2,9), NE);
 +
 +
draw((15,3)--(16,3)--(16,5)--(15,5)--cycle,linewidth(1.125));
 +
dot((15,3),linewidth(4));
 +
dot((16,3),linewidth(4));
 +
dot((16,5),linewidth(4));
 +
dot((15,5),linewidth(4));
 +
</asy>
 
<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 85: Line 334:
  
 
==Problem 9==  
 
==Problem 9==  
−
Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between 4 and 7 meters?
+
Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between <math>4</math> and <math>7</math> meters?
−
 
+
<asy>
 +
// Diagram by TheMathGuyd. Found cubic, so graph is perfect.
 +
import graph;
 +
size(8cm);
 +
int i;
 +
for(i=1; i<9; i=i+1)
 +
{
 +
draw((-0.2,2i-1)--(16.2,2i-1), mediumgrey);
 +
draw((2i-1,-0.2)--(2i-1,16.2), mediumgrey);
 +
draw((-0.2,2i)--(16.2,2i), grey);
 +
draw((2i,-0.2)--(2i,16.2), grey);
 +
}
 +
Label f;
 +
f.p=fontsize(6);
 +
xaxis(-0.5,17.8,Ticks(f, 2.0),Arrow());
 +
yaxis(-0.5,17.8,Ticks(f, 2.0),Arrow());
 +
real f(real x)
 +
{
 +
return -0.03125 x^(3) + 0.75x^(2) - 5.125 x + 14.5;
 +
}
 +
draw(graph(f,0,15.225),currentpen+1);
 +
real dpt=2;
 +
real ts=0.75;
 +
transform st=scale(ts);
 +
label(rotate(90)*st*"Elevation (meters)",(-dpt,8));
 +
label(st*"Time (seconds)",(8,-dpt));
 +
</asy>
 
<math>\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math>
 
<math>\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math>
  
 +
[[2023 AMC 8 Problems/Problem 9|Solution]]
  
−
[[2023 AMC 8 Problems/Problem 9|Solution]]
 
 
==Problem 10==
 
==Problem 10==
  
Line 111: Line 386:
 
interior. What fraction of the interior of the large white circle is shaded?
 
interior. What fraction of the interior of the large white circle is shaded?
  
−
[[Image:2023 AMC 8-12.png|thumb|center|300px]]
+
<asy>
 +
// Diagram by TheMathGuyd
 +
size(6cm);
 +
draw(circle((3,3),3));
 +
filldraw(circle((2,3),2),lightgrey);
 +
filldraw(circle((3,3),1),white);
 +
filldraw(circle((1,3),1),white);
 +
filldraw(circle((5.5,3),0.5),lightgrey);
 +
filldraw(circle((4.5,4.5),0.5),lightgrey);
 +
filldraw(circle((4.5,1.5),0.5),lightgrey);
 +
int i, j;
 +
for(i=0; i<7; i=i+1)
 +
{
 +
draw((0,i)--(6,i), dashed+grey);
 +
draw((i,0)--(i,6), dashed+grey);
 +
}
 +
</asy>
  
 
<math>\textbf{(A)}\ \frac{1}{4} \qquad \textbf{(B)}\ \frac{11}{36} \qquad \textbf{(C)}\ \frac{1}{3} \qquad \textbf{(D)}\ \frac{19}{36} \qquad \textbf{(E)}\ \frac{5}{9}</math>
 
<math>\textbf{(A)}\ \frac{1}{4} \qquad \textbf{(B)}\ \frac{11}{36} \qquad \textbf{(C)}\ \frac{1}{3} \qquad \textbf{(D)}\ \frac{19}{36} \qquad \textbf{(E)}\ \frac{5}{9}</math>
 +
  
 
[[2023 AMC 8 Problems/Problem 12|Solution]]
 
[[2023 AMC 8 Problems/Problem 12|Solution]]
Line 119: Line 411:
 
==Problem 13==
 
==Problem 13==
  
−
Along the route of a bicycle race, 7 water stations are evenly spaced between the start and finish lines,
+
Along the route of a bicycle race, <math>7</math> water stations are evenly spaced between the start and finish lines,
−
as shown in the figure below. There are also 2 repair stations evenly spaced between the start and
+
as shown in the figure below. There are also <math>2</math> repair stations evenly spaced between the start and
−
finish lines. The 3rd water station is located 2 miles after the 1st repair station. How long is the race
+
finish lines. The <math>3</math>rd water station is located <math>2</math> miles after the <math>1</math>st repair station. How long is the race
 
in miles?
 
in miles?
 +
<asy>
 +
// Credits given to Themathguyd‎ and Kante314
 +
usepackage("mathptmx");
 +
size(10cm);
 +
filldraw((11,4.5)--(171,4.5)--(171,17.5)--(11,17.5)--cycle,mediumgray*0.4 + lightgray*0.6);
 +
draw((11,11)--(171,11),linetype("2 2")+white+linewidth(1.2));
 +
draw((0,0)--(11,0)--(11,22)--(0,22)--cycle);
 +
draw((171,0)--(182,0)--(182,22)--(171,22)--cycle);
  
 +
draw((31,4.5)--(31,0));
 +
draw((51,4.5)--(51,0));
 +
draw((151,4.5)--(151,0));
 +
 +
label(scale(.85)*rotate(45)*"Water 1", (23,-13.5));
 +
label(scale(.85)*rotate(45)*"Water 2", (43,-13.5));
 +
label(scale(.85)*rotate(45)*"Water 7", (143,-13.5));
 +
 +
filldraw(circle((103,-13.5),.2));
 +
filldraw(circle((98,-13.5),.2));
 +
filldraw(circle((93,-13.5),.2));
 +
filldraw(circle((88,-13.5),.2));
 +
filldraw(circle((83,-13.5),.2));
 +
 +
label(scale(.85)*rotate(90)*"Start", (5.5,11));
 +
label(scale(.85)*rotate(270)*"Finish", (176.5,11));
 +
</asy>
 
<math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 96</math>
 
<math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 96</math>
  
Line 129: Line 446:
  
 
==Problem 14==
 
==Problem 14==
−
Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of 5-cent, 10-cent, and 25-cent stamps, with exactly 20 of each type. What is the greatest number of stamps Nicolas can use to make exactly \$7.10 in postage?
+
Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of <math>5</math>-cent, <math>10</math>-cent, and <math>25</math>-cent stamps, with exactly <math>20</math> of each type. What is the greatest number of stamps Nicolas can use to make exactly <math>\$7.10</math> in postage?
−
(Note: The amount \$7.10 corresponds to 7 dollars and 10 cents. One dollar is worth 100 cents.)
+
(Note: The amount <math>\$7.10</math> corresponds to <math>7</math> dollars and <math>10</math> cents. One dollar is worth <math>100</math> cents.)
  
 
<math>\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 54\qquad \textbf{(E)}\ 55</math>
 
<math>\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 54\qquad \textbf{(E)}\ 55</math>
Line 138: Line 455:
 
==Problem 15==
 
==Problem 15==
  
−
Viswam walks half a mile to get to school each day. His route consists of 10 city blocks of equal length and he takes 1 minute to walk each block. Today, after walking 5 blocks, Viswam discovers he has to make a detour, walking 3 blocks of equal length instead of 1 block to reach the next corner. From the time he starts his detour, at what speed, in mph, must he walk, in order to get to school at his usual time?
+
Viswam walks half a mile to get to school each day. His route consists of <math>10</math> city blocks of equal length and he takes <math>1</math> minute to walk each block. Today, after walking <math>5</math> blocks, Viswam discovers he has to make a detour, walking <math>3</math> blocks of equal length instead of <math>1</math> block to reach the next corner. From the time he starts his detour, at what speed, in mph, must he walk, in order to get to school at his usual time?
 
<asy>
 
<asy>
 
// Diagram by TheMathGuyd
 
// Diagram by TheMathGuyd
−
size(8cm);
+
size(13cm);
 
// this is an important stickman to the left of the origin
 
// this is an important stickman to the left of the origin
 
pair C=midpoint((-0.5,0.5)--(-0.6,0.05));
 
pair C=midpoint((-0.5,0.5)--(-0.6,0.05));
Line 163: Line 480:
 
fill((5+d,-d/2)--(6-d,-d/2)--(6-d,d/2)--(5+d,d/2)--cycle,lightred);
 
fill((5+d,-d/2)--(6-d,-d/2)--(6-d,d/2)--(5+d,d/2)--cycle,lightred);
  
−
draw((0,0)--(5,0)--(5,1)--(6,1)--(6,0)--(10.1,0),black);
+
draw((0,0)--(5,0)--(5,1)--(6,1)--(6,0)--(10.1,0),deepblue+linewidth(1.25)); //Who even noticed
 
label("School", (10,0),E, Draw());
 
label("School", (10,0),E, Draw());
 
</asy>
 
</asy>
Line 171: Line 488:
  
 
==Problem 16==
 
==Problem 16==
−
The letters P, Q, and R are entered into a <math>20\times20</math> table according to the pattern shown below. How many Ps, Qs, and Rs will appear in the completed table?  
+
The letters <math>\text{P}, \text{Q},</math> and <math>\text{R}</math> are entered into a <math>20\times20</math> table according to the pattern shown below. How many <math>\text{P}</math>s, <math>\text{Q}</math>s, and <math>\text{R}</math>s will appear in the completed table?  
 +
<asy>
 +
/* Made by MRENTHUSIASM, Edited by Kante314 */
 +
usepackage("mathdots");
 +
size(5cm);
 +
draw((0,0)--(6,0),linewidth(1.5)+mediumgray);
 +
draw((0,1)--(6,1),linewidth(1.5)+mediumgray);
 +
draw((0,2)--(6,2),linewidth(1.5)+mediumgray);
 +
draw((0,3)--(6,3),linewidth(1.5)+mediumgray);
 +
draw((0,4)--(6,4),linewidth(1.5)+mediumgray);
 +
draw((0,5)--(6,5),linewidth(1.5)+mediumgray);
 +
 
 +
draw((0,0)--(0,6),linewidth(1.5)+mediumgray);
 +
draw((1,0)--(1,6),linewidth(1.5)+mediumgray);
 +
draw((2,0)--(2,6),linewidth(1.5)+mediumgray);
 +
draw((3,0)--(3,6),linewidth(1.5)+mediumgray);
 +
draw((4,0)--(4,6),linewidth(1.5)+mediumgray);
 +
draw((5,0)--(5,6),linewidth(1.5)+mediumgray);
 +
 
 +
label(scale(.9)*"\textsf{P}", (.5,.5));
 +
label(scale(.9)*"\textsf{Q}", (.5,1.5));
 +
label(scale(.9)*"\textsf{R}", (.5,2.5));
 +
label(scale(.9)*"\textsf{P}", (.5,3.5));
 +
label(scale(.9)*"\textsf{Q}", (.5,4.5));
 +
label("$\vdots$", (.5,5.6));
 +
 
 +
label(scale(.9)*"\textsf{Q}", (1.5,.5));
 +
label(scale(.9)*"\textsf{R}", (1.5,1.5));
 +
label(scale(.9)*"\textsf{P}", (1.5,2.5));
 +
label(scale(.9)*"\textsf{Q}", (1.5,3.5));
 +
label(scale(.9)*"\textsf{R}", (1.5,4.5));
 +
label("$\vdots$", (1.5,5.6));
 +
 
 +
label(scale(.9)*"\textsf{R}", (2.5,.5));
 +
label(scale(.9)*"\textsf{P}", (2.5,1.5));
 +
label(scale(.9)*"\textsf{Q}", (2.5,2.5));
 +
label(scale(.9)*"\textsf{R}", (2.5,3.5));
 +
label(scale(.9)*"\textsf{P}", (2.5,4.5));
 +
label("$\vdots$", (2.5,5.6));
  
−
<cmath>
+
label(scale(.9)*"\textsf{P}", (3.5,.5));
−
%%Table made by Technodoggo
+
label(scale(.9)*"\textsf{Q}", (3.5,1.5));
−
\begin{array}[b]{|c|c|c|c|c|c}
+
label(scale(.9)*"\textsf{R}", (3.5,2.5));
−
\vdots &\vdots&\vdots&\vdots&\vdots&\iddots\\\hline
+
label(scale(.9)*"\textsf{P}", (3.5,3.5));
−
Q&R&P&Q&R&\cdots\\\hline
+
label(scale(.9)*"\textsf{Q}", (3.5,4.5));
−
P&Q&R&P&Q&\cdots\\\hline
+
label("$\vdots$", (3.5,5.6));
−
R&P&Q&R&P&\cdots\\\hline
 
−
Q&R&P&Q&R&\cdots\\\hline
 
−
P&Q&R&P&Q&\cdots\\\hline
 
−
\end{array}</cmath>
 
  
 +
label(scale(.9)*"\textsf{Q}", (4.5,.5));
 +
label(scale(.9)*"\textsf{R}", (4.5,1.5));
 +
label(scale(.9)*"\textsf{P}", (4.5,2.5));
 +
label(scale(.9)*"\textsf{Q}", (4.5,3.5));
 +
label(scale(.9)*"\textsf{R}", (4.5,4.5));
 +
label("$\vdots$", (4.5,5.6));
  
 +
label(scale(.9)*"$\dots$", (5.5,.5));
 +
label(scale(.9)*"$\dots$", (5.5,1.5));
 +
label(scale(.9)*"$\dots$", (5.5,2.5));
 +
label(scale(.9)*"$\dots$", (5.5,3.5));
 +
label(scale(.9)*"$\dots$", (5.5,4.5));
 +
label(scale(.9)*"$\iddots$", (5.5,5.6));
 +
</asy>
 
<math>\textbf{(A)}~132\text{ Ps, }134\text{ Qs, }134\text{ Rs}</math>  
 
<math>\textbf{(A)}~132\text{ Ps, }134\text{ Qs, }134\text{ Rs}</math>  
  
Line 198: Line 562:
  
 
==Problem 17==
 
==Problem 17==
−
A ''regular octahedron'' has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of <math>Q</math>?
+
A <i>regular octahedron</i> has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of <math>Q</math>?
 +
 
 +
<asy>
 +
// Diagram by TheMathGuyd
 +
import graph;
 +
// The Solid
 +
// To save processing time, do not use three (dimensions)
 +
// Project (roughly) to two
 +
size(15cm);
 +
pair Fr, Lf, Rt, Tp, Bt, Bk;
 +
Lf=(0,0);
 +
Rt=(12,1);
 +
Fr=(7,-1);
 +
Bk=(5,2);
 +
Tp=(6,6.7);
 +
Bt=(6,-5.2);
 +
draw(Lf--Fr--Rt);
 +
draw(Lf--Tp--Rt);
 +
draw(Lf--Bt--Rt);
 +
draw(Tp--Fr--Bt);
 +
draw(Lf--Bk--Rt,dashed);
 +
draw(Tp--Bk--Bt,dashed);
 +
label(rotate(-8.13010235)*slant(0.1)*"$Q$", (4.2,1.6));
 +
label(rotate(21.8014095)*slant(-0.2)*"$?$", (8.5,2.05));
 +
pair g = (-8,0); // Define Gap transform
 +
real a = 8;
 +
draw(g+(-a/2,1)--g+(a/2,1), Arrow()); // Make arrow
 +
// Time for the NET
 +
pair DA,DB,DC,CD,O;
 +
DA = (4*sqrt(3),0);
 +
DB = (2*sqrt(3),6);
 +
DC = (DA+DB)/3;
 +
CD = conj(DC);
 +
O=(0,0);
 +
transform trf=shift(3g+(0,3));
 +
path NET = O--(-2*DA)--(-2DB)--(-DB)--(2DA-DB)--DB--O--DA--(DA-DB)--O--(-DB)--(-DA)--(-DA-DB)--(-DB);
 +
draw(trf*NET);
 +
label("$7$",trf*DC);
 +
label("$Q$",trf*DC+DA-DB);
 +
label("$5$",trf*DC-DB);
 +
label("$3$",trf*DC-DA-DB);
 +
label("$6$",trf*CD);
 +
label("$4$",trf*CD-DA);
 +
label("$2$",trf*CD-DA-DB);
 +
label("$1$",trf*CD-2DA);
 +
</asy>
  
 
<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5</math>
 
<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5</math>
Line 213: Line 622:
  
 
==Problem 19==
 
==Problem 19==
−
An equilateral triangle is placed inside a larger equilateral triangle so that the region between them
+
An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is <math>\frac23</math> the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?
−
can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle
 
−
is <math>\frac{2}{3}</math> the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of
 
−
the inner triangle?
 
  
 
<asy>
 
<asy>
Line 238: Line 644:
 
</asy>
 
</asy>
  
−
<math>\textbf{(A)}\ 1:3 \qquad \textbf{(B)}\ 3:8 \qquad \textbf{(C)}\ 5:12 \qquad \textbf{(D)}\ 7:16 \qquad \textbf{(E)}\ 4:9</math>
+
<math>\textbf{(A) } 1 : 3 \qquad \textbf{(B) } 3 : 8 \qquad \textbf{(C) } 5 : 12 \qquad \textbf{(D) } 7 : 16 \qquad \textbf{(E) } 4 : 9</math>
  
 
[[2023 AMC 8 Problems/Problem 19|Solution]]
 
[[2023 AMC 8 Problems/Problem 19|Solution]]
Line 244: Line 650:
 
==Problem 20==
 
==Problem 20==
  
−
Two integers are inserted into the list <math>3,3,8,11,28</math> to double its range. The mode and median remain unchanged. What is the maximum possible sum of two additional numbers?
+
Two integers are inserted into the list <math>3,3,8,11,28</math> to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?
  
 
<math>\textbf{(A)}\ 56 \qquad \textbf{(B)}\ 57 \qquad \textbf{(C)}\ 58 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 61</math>
 
<math>\textbf{(A)}\ 56 \qquad \textbf{(B)}\ 57 \qquad \textbf{(C)}\ 58 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 61</math>
Line 251: Line 657:
  
 
==Problem 21==
 
==Problem 21==
−
Alina writes the numbers <math>1, 2, \dots , 9</math> on separate cards, one number per card. She wishes to divide the cards into <math>3</math> groups of <math>3</math> cards so that the sum of the number in each group will be the same. In how many ways can this be done?
+
Alina writes the numbers <math>1, 2, \dots , 9</math> on separate cards, one number per card. She wishes to divide the cards into <math>3</math> groups of <math>3</math> cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?
  
 
<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 258: Line 664:
  
 
==Problem 22==
 
==Problem 22==
−
 
 
In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is <math>4000</math>. What is the first term?
 
In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is <math>4000</math>. What is the first term?
  
Line 266: Line 671:
  
 
==Problem 23==
 
==Problem 23==
−
Each square in a <math>3 \times 3</math> grid is randomly filled with one of the <math>4</math> gray and white tiles shown below on the right.
+
Each square in a <math>3 \times 3</math> grid is randomly filled with one of the <math>4</math> gray and white tiles shown below on the right.  
 +
<asy>
 +
size(5.663333333cm);
 +
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,gray);
 +
draw((1,0)--(1,3)--(2,3)--(2,0),gray);
 +
draw((0,1)--(3,1)--(3,2)--(0,2),gray);
  
−
<asy> size(5.663333333cm); draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,gray); draw((1,0)--(1,3)--(2,3)--(2,0),gray); draw((0,1)--(3,1)--(3,2)--(0,2),gray);  fill((6,.33)--(7,.33)--(7,1.33)--cycle,mediumgray); draw((6,.33)--(7,.33)--(7,1.33)--(6,1.33)--cycle,gray); fill((6,1.67)--(7,2.67)--(6,2.67)--cycle,mediumgray); draw((6,1.67)--(7,1.67)--(7,2.67)--(6,2.67)--cycle,gray); fill((7.33,.33)--(8.33,.33)--(7.33,1.33)--cycle,mediumgray); draw((7.33,.33)--(8.33,.33)--(8.33,1.33)--(7.33,1.33)--cycle,gray); fill((8.33,1.67)--(8.33,2.67)--(7.33,2.67)--cycle,mediumgray); draw((7.33,1.67)--(8.33,1.67)--(8.33,2.67)--(7.33,2.67)--cycle,gray); size(2cm);  fill((1,0)--(0,1)--(0,2)--(1,1)--cycle,mediumgray); fill((2,0)--(3,1)--(2,2)--(1,1)--cycle,mediumgray); fill((1,2)--(1,3)--(0,3)--cycle,mediumgray); fill((1,2)--(2,2)--(2,3)--cycle,mediumgray); fill((3,2)--(3,3)--(2,3)--cycle,mediumgray);  draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,gray); draw((1,0)--(1,3)--(2,3)--(2,0),gray); draw((0,1)--(3,1)--(3,2)--(0,2),gray); </asy>
+
fill((6,.33)--(7,.33)--(7,1.33)--cycle,mediumgray);
 +
draw((6,.33)--(7,.33)--(7,1.33)--(6,1.33)--cycle,gray);
 +
fill((6,1.67)--(7,2.67)--(6,2.67)--cycle,mediumgray);
 +
draw((6,1.67)--(7,1.67)--(7,2.67)--(6,2.67)--cycle,gray);
 +
fill((7.33,.33)--(8.33,.33)--(7.33,1.33)--cycle,mediumgray);
 +
draw((7.33,.33)--(8.33,.33)--(8.33,1.33)--(7.33,1.33)--cycle,gray);
 +
fill((8.33,1.67)--(8.33,2.67)--(7.33,2.67)--cycle,mediumgray);
 +
draw((7.33,1.67)--(8.33,1.67)--(8.33,2.67)--(7.33,2.67)--cycle,gray);
 +
</asy>
 +
What is the probability that the tiling will contain a large gray diamond in one of the smaller <math>2 \times 2</math> grids? Below is an example of such tiling.
 +
<asy>
 +
size(2cm);
  
−
What is the probability that the tiling will contain a large gray diamond in one of the smaller <math>2 \times 2</math> grids? Below is an example of such tiling.
+
fill((1,0)--(0,1)--(0,2)--(1,1)--cycle,mediumgray);
 +
fill((2,0)--(3,1)--(2,2)--(1,1)--cycle,mediumgray);
 +
fill((1,2)--(1,3)--(0,3)--cycle,mediumgray);
 +
fill((1,2)--(2,2)--(2,3)--cycle,mediumgray);
 +
fill((3,2)--(3,3)--(2,3)--cycle,mediumgray);
 +
 
 +
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,gray);
 +
draw((1,0)--(1,3)--(2,3)--(2,0),gray);
 +
draw((0,1)--(3,1)--(3,2)--(0,2),gray);
 +
</asy>
  
 
<math>\textbf{(A) } \frac{1}{1024} \qquad \textbf{(B) } \frac{1}{256} \qquad \textbf{(C) } \frac{1}{64} \qquad \textbf{(D) } \frac{1}{16} \qquad \textbf{(E) } \frac{1}{4}</math>
 
<math>\textbf{(A) } \frac{1}{1024} \qquad \textbf{(B) } \frac{1}{256} \qquad \textbf{(C) } \frac{1}{64} \qquad \textbf{(D) } \frac{1}{16} \qquad \textbf{(E) } \frac{1}{4}</math>
Line 277: Line 707:
  
 
==Problem 24==
 
==Problem 24==
−
Isosceles triangle <math>ABC</math> has equal side lengths <math>AB</math> and <math>BC</math>. In the figures below, segments are drawn parallel to <math>\overline{AC}</math> so that the shaded portions of <math>\triangle</math><math>ABC</math> have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height <math>h</math> of <math>\triangle</math><math>ABC</math>?
+
Isosceles triangle <math>ABC</math> has equal side lengths <math>AB</math> and <math>BC</math>. In the figures below, segments are drawn parallel to <math>\overline{AC}</math> so that the shaded portions of <math>\triangle ABC</math> have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height <math>h</math> of <math>\triangle ABC</math>?
 +
 
 +
<asy>
 +
//Diagram by TheMathGuyd
 +
size(12cm);
 +
real h = 2.5; // height
 +
real g=4; //c2c space
 +
real s = 0.65; //Xcord of Hline
 +
real adj = 0.08; //adjust line diffs
 +
pair A,B,C;
 +
B=(0,h);
 +
C=(1,0);
 +
A=-conj(C);
 +
pair PONE=(s,h*(1-s)); //Endpoint of Hline ONE
 +
pair PTWO=(s+adj,h*(1-s-adj)); //Endpoint of Hline ONE
 +
path LONE=PONE--(-conj(PONE)); //Hline ONE
 +
path LTWO=PTWO--(-conj(PTWO));
 +
path T=A--B--C--cycle; //Triangle
 +
 
 +
 
 +
fill (shift(g,0)*(LTWO--B--cycle),mediumgrey);
 +
fill (LONE--A--C--cycle,mediumgrey);
 +
 
 +
draw(LONE);
 +
draw(T);
 +
label("$A$",A,SW);
 +
label("$B$",B,N);
 +
label("$C$",C,SE);
 +
 
 +
draw(shift(g,0)*LTWO);
 +
draw(shift(g,0)*T);
 +
label("$A$",shift(g,0)*A,SW);
 +
label("$B$",shift(g,0)*B,N);
 +
label("$C$",shift(g,0)*C,SE);
 +
 
 +
draw(B--shift(g,0)*B,dashed);
 +
draw(C--shift(g,0)*A,dashed);
 +
draw((g/2,0)--(g/2,h),dashed);
 +
draw((0,h*(1-s))--B,dashed);
 +
draw((g,h*(1-s-adj))--(g,0),dashed);
 +
label("$5$", midpoint((g,h*(1-s-adj))--(g,0)),UnFill);
 +
label("$h$", midpoint((g/2,0)--(g/2,h)),UnFill);
 +
label("$11$", midpoint((0,h*(1-s))--B),UnFill);
 +
</asy>
  
 
<math>\textbf{(A)}\ 14.6 \qquad \textbf{(B)}\ 14.8 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 15.2 \qquad \textbf{(E)}\ 15.4</math>
 
<math>\textbf{(A)}\ 14.6 \qquad \textbf{(B)}\ 14.8 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 15.2 \qquad \textbf{(E)}\ 15.4</math>
Line 286: Line 759:
  
 
Fifteen integers <math>a_1, a_2, a_3, \dots, a_{15}</math> are arranged in order on a number line. The integers are equally spaced and have the property that
 
Fifteen integers <math>a_1, a_2, a_3, \dots, a_{15}</math> are arranged in order on a number line. The integers are equally spaced and have the property that
−
<cmath>1 \le a_1 \le 10, \thickspace 13 \le a_2 \le 20, \thickspace 241 \le a_{15}\le 250.</cmath>
+
<cmath>1 \le a_1 \le 10, \thickspace 13 \le a_2 \le 20, \thickspace \text{ and } \thickspace 241 \le a_{15}\le 250.</cmath>
−
What is the sum of digits of <math>a_{14}</math>?
+
What is the sum of digits of <math>a_{14}?</math>
  
 
<math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 12</math>
 
<math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 12</math>

Latest revision as of 20:26, 1 October 2024

2023 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

What is the value of $(8 \times 4 + 2) - (8 + 4 \times 2)$?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$

Solution

Problem 2

A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures? [asy] //Restored original diagram. Alter it if you would like, but it was made by TheMathGuyd, // Diagram by TheMathGuyd. I even put the lined texture :) // Thank you Kante314 for inspiring thicker arrows. They do look much better size(0,3cm); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle; path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; filldraw(sq,mediumgrey,black); draw((0.75,0)--(1.25,0),currentpen+1,Arrow(size=6)); //folding path sqside = (-0.5,-0.5)--(0.5,-0.5); path rhside = (-0.125,-0.125)--(0.5,-0.5); transform fld = shift((1.75,0))*scale(0.5); draw(fld*sq,black); int i; for(i=0; i<10; i=i+1) { draw(shift(0,0.05*i)*fld*sqside,deepblue); } path rhedge = (-0.125,-0.125)--(-0.125,0.8)--(-0.2,0.85)--cycle; filldraw(fld*rhedge,grey); path sqedge = (-0.5,-0.5)--(-0.5,0.4475)--(-0.575,0.45)--cycle; filldraw(fld*sqedge,grey); filldraw(fld*rh,white,black); int i; for(i=0; i<10; i=i+1) { draw(shift(0,0.05*i)*fld*rhside,deepblue); } draw((2.25,0)--(2.75,0),currentpen+1,Arrow(size=6)); //cutting transform cut = shift((3.25,0))*scale(0.5); draw(shift((-0.01,+0.01))*cut*sq); draw(cut*sq); filldraw(shift((0.01,-0.01))*cut*sq,white,black); int j; for(j=0; j<10; j=j+1) { draw(shift(0,0.05*j)*cut*sqside,deepblue); } draw(shift((0.01,-0.01))*cut*(0,-0.5)--shift((0.01,-0.01))*cut*(0.5,0),dashed); //Answers Below, but already Separated //filldraw(sqA,grey,black); //filldraw(sqB,grey,black); //filldraw(sq,grey,black); //filldraw(sqC,white,black); //filldraw(sq,grey,black); //filldraw(trD,white,black); //filldraw(sq,grey,black); //filldraw(sqE,white,black); [/asy]

[asy] // Diagram by TheMathGuyd. size(0,7.5cm); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle; path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; //ANSWERS real sh = 1.5; label("$\textbf{(A)}$",(-0.5,0.5),SW); label("$\textbf{(B)}$",shift((sh,0))*(-0.5,0.5),SW); label("$\textbf{(C)}$",shift((2sh,0))*(-0.5,0.5),SW); label("$\textbf{(D)}$",shift((0,-sh))*(-0.5,0.5),SW); label("$\textbf{(E)}$",shift((sh,-sh))*(-0.5,0.5),SW); filldraw(sqA,mediumgrey,black); filldraw(shift((sh,0))*sqB,mediumgrey,black); filldraw(shift((2*sh,0))*sq,mediumgrey,black); filldraw(shift((2*sh,0))*sqC,white,black); filldraw(shift((0,-sh))*sq,mediumgrey,black); filldraw(shift((0,-sh))*trD,white,black); filldraw(shift((sh,-sh))*sq,mediumgrey,black); filldraw(shift((sh,-sh))*sqE,white,black); [/asy]

Solution

Problem 3

Wind chill is a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be found using this calculation \[(\text{wind chill}) = (\text{air temperature}) - 0.7 \times (\text{wind speed}),\] where temperature is measured in degrees Fahrenheit $(^{\circ}\text{F})$ and the wind speed is measured in miles per hour (mph). Suppose the air temperature is $36^{\circ}\text{F}$ and the wind speed is $18$ mph. Which of the following is closest to the approximate wind chill?

$\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 35$

Solution

Problem 4

The numbers from $1$ to $49$ are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number $7.$ How many of these four numbers are prime? [asy] /* Made by MRENTHUSIASM */ size(175); void ds(pair p) { filldraw((0.5,0.5)+p--(-0.5,0.5)+p--(-0.5,-0.5)+p--(0.5,-0.5)+p--cycle,mediumgrey); } ds((0.5,4.5)); ds((1.5,3.5)); ds((3.5,1.5)); ds((4.5,0.5)); add(grid(7,7,grey+linewidth(1.25))); int adj = 1; int curUp = 2; int curLeft = 4; int curDown = 6; label("$1$",(3.5,3.5)); for (int len = 3; len<=3; len+=2) { for (int i=1; i<=len-1; ++i) { label("$"+string(curUp)+"$",(3.5+adj,3.5-adj+i)); label("$"+string(curLeft)+"$",(3.5+adj-i,3.5+adj)); label("$"+string(curDown)+"$",(3.5-adj,3.5+adj-i)); ++curDown; ++curLeft; ++curUp; } ++adj; curUp = len^2 + 1; curLeft = len^2 + len + 2; curDown = len^2 + 2*len + 3; } draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(2)); [/asy] $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Solution

Problem 5

A lake contains $250$ trout, along with a variety of other fish. When a marine biologist catches and releases a sample of $180$ fish from the lake, $30$ are identified as trout. Assume that the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake?

$\textbf{(A)}\ 1250 \qquad \textbf{(B)}\ 1500 \qquad \textbf{(C)}\ 1750 \qquad \textbf{(D)}\ 1800 \qquad \textbf{(E)}\ 2000$

Solution

Problem 6

The digits $2, 0, 2,$ and $3$ are placed in the expression below, one digit per box. What is the maximum possible value of the expression?

[asy] // Diagram by TheMathGuyd. I can compress this later size(5cm); real w=2.2; pair O,I,J; O=(0,0);I=(1,0);J=(0,1); path bsqb = O--I; path bsqr = I--I+J; path bsqt = I+J--J; path bsql = J--O; path lsqb = shift((1.2,0.75))*scale(0.5)*bsqb; path lsqr = shift((1.2,0.75))*scale(0.5)*bsqr; path lsqt = shift((1.2,0.75))*scale(0.5)*bsqt; path lsql = shift((1.2,0.75))*scale(0.5)*bsql; draw(bsqb,dashed); draw(bsqr,dashed); draw(bsqt,dashed); draw(bsql,dashed); draw(lsqb,dashed); draw(lsqr,dashed); draw(lsqt,dashed); draw(lsql,dashed); label(scale(3)*"$\times$",(w,1/3)); draw(shift(1.3w,0)*bsqb,dashed); draw(shift(1.3w,0)*bsqr,dashed); draw(shift(1.3w,0)*bsqt,dashed); draw(shift(1.3w,0)*bsql,dashed); draw(shift(1.3w,0)*lsqb,dashed); draw(shift(1.3w,0)*lsqr,dashed); draw(shift(1.3w,0)*lsqt,dashed); draw(shift(1.3w,0)*lsql,dashed); [/asy]

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 18$

Solution

Problem 7

A rectangle, with sides parallel to the $x$-axis and $y$-axis, has opposite vertices located at $(15, 3)$ and $(16, 5)$. A line is drawn through points $A(0, 0)$ and $B(3, 1)$. Another line is drawn through points $C(0, 10)$ and $D(2, 9)$. How many points on the rectangle lie on at least one of the two lines? [asy] usepackage("mathptmx"); size(9cm); draw((0,-.5)--(0,11),EndArrow(size=.15cm)); draw((1,0)--(1,11),mediumgray); draw((2,0)--(2,11),mediumgray); draw((3,0)--(3,11),mediumgray); draw((4,0)--(4,11),mediumgray); draw((5,0)--(5,11),mediumgray); draw((6,0)--(6,11),mediumgray); draw((7,0)--(7,11),mediumgray); draw((8,0)--(8,11),mediumgray); draw((9,0)--(9,11),mediumgray); draw((10,0)--(10,11),mediumgray); draw((11,0)--(11,11),mediumgray); draw((12,0)--(12,11),mediumgray); draw((13,0)--(13,11),mediumgray); draw((14,0)--(14,11),mediumgray); draw((15,0)--(15,11),mediumgray); draw((16,0)--(16,11),mediumgray); draw((-.5,0)--(17,0),EndArrow(size=.15cm)); draw((0,1)--(17,1),mediumgray); draw((0,2)--(17,2),mediumgray); draw((0,3)--(17,3),mediumgray); draw((0,4)--(17,4),mediumgray); draw((0,5)--(17,5),mediumgray); draw((0,6)--(17,6),mediumgray); draw((0,7)--(17,7),mediumgray); draw((0,8)--(17,8),mediumgray); draw((0,9)--(17,9),mediumgray); draw((0,10)--(17,10),mediumgray); draw((-.13,1)--(.13,1)); draw((-.13,2)--(.13,2)); draw((-.13,3)--(.13,3)); draw((-.13,4)--(.13,4)); draw((-.13,5)--(.13,5)); draw((-.13,6)--(.13,6)); draw((-.13,7)--(.13,7)); draw((-.13,8)--(.13,8)); draw((-.13,9)--(.13,9)); draw((-.13,10)--(.13,10)); draw((1,-.13)--(1,.13)); draw((2,-.13)--(2,.13)); draw((3,-.13)--(3,.13)); draw((4,-.13)--(4,.13)); draw((5,-.13)--(5,.13)); draw((6,-.13)--(6,.13)); draw((7,-.13)--(7,.13)); draw((8,-.13)--(8,.13)); draw((9,-.13)--(9,.13)); draw((10,-.13)--(10,.13)); draw((11,-.13)--(11,.13)); draw((12,-.13)--(12,.13)); draw((13,-.13)--(13,.13)); draw((14,-.13)--(14,.13)); draw((15,-.13)--(15,.13)); draw((16,-.13)--(16,.13)); label(scale(.7)*"$1$", (1,-.13), S); label(scale(.7)*"$2$", (2,-.13), S); label(scale(.7)*"$3$", (3,-.13), S); label(scale(.7)*"$4$", (4,-.13), S); label(scale(.7)*"$5$", (5,-.13), S); label(scale(.7)*"$6$", (6,-.13), S); label(scale(.7)*"$7$", (7,-.13), S); label(scale(.7)*"$8$", (8,-.13), S); label(scale(.7)*"$9$", (9,-.13), S); label(scale(.7)*"$10$", (10,-.13), S); label(scale(.7)*"$11$", (11,-.13), S); label(scale(.7)*"$12$", (12,-.13), S); label(scale(.7)*"$13$", (13,-.13), S); label(scale(.7)*"$14$", (14,-.13), S); label(scale(.7)*"$15$", (15,-.13), S); label(scale(.7)*"$16$", (16,-.13), S); label(scale(.7)*"$1$", (-.13,1), W); label(scale(.7)*"$2$", (-.13,2), W); label(scale(.7)*"$3$", (-.13,3), W); label(scale(.7)*"$4$", (-.13,4), W); label(scale(.7)*"$5$", (-.13,5), W); label(scale(.7)*"$6$", (-.13,6), W); label(scale(.7)*"$7$", (-.13,7), W); label(scale(.7)*"$8$", (-.13,8), W); label(scale(.7)*"$9$", (-.13,9), W); label(scale(.7)*"$10$", (-.13,10), W); dot((0,0),linewidth(4)); label(scale(.75)*"$A$", (0,0), NE); dot((3,1),linewidth(4)); label(scale(.75)*"$B$", (3,1), NE); dot((0,10),linewidth(4)); label(scale(.75)*"$C$", (0,10), NE); dot((2,9),linewidth(4)); label(scale(.75)*"$D$", (2,9), NE); draw((15,3)--(16,3)--(16,5)--(15,5)--cycle,linewidth(1.125)); dot((15,3),linewidth(4)); dot((16,3),linewidth(4)); dot((16,5),linewidth(4)); dot((15,5),linewidth(4)); [/asy] $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Solution

Problem 8

Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against each of the other three players exactly twice. Shown below are the win-loss records for the players. The numbers $1$ and $0$ represent a win or loss, respectively. For example, Lola won five matches and lost the fourth match. What was Tiyo’s win-loss record?

\[\begin{tabular}{c | c} Player & Result \\ \hline Lola & \texttt{111011}\\ Lolo & \texttt{101010}\\ Tiya & \texttt{010100}\\ Tiyo & \texttt{??????} \end{tabular}\]

$\textbf{(A)}\ \texttt{000101} \qquad \textbf{(B)}\ \texttt{001001} \qquad \textbf{(C)}\ \texttt{010000} \qquad \textbf{(D)}\ \texttt{010101} \qquad \textbf{(E)}\ \texttt{011000}$

Solution

Problem 9

Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between $4$ and $7$ meters? [asy] // Diagram by TheMathGuyd. Found cubic, so graph is perfect. import graph; size(8cm); int i; for(i=1; i<9; i=i+1) { draw((-0.2,2i-1)--(16.2,2i-1), mediumgrey); draw((2i-1,-0.2)--(2i-1,16.2), mediumgrey); draw((-0.2,2i)--(16.2,2i), grey); draw((2i,-0.2)--(2i,16.2), grey); } Label f; f.p=fontsize(6); xaxis(-0.5,17.8,Ticks(f, 2.0),Arrow()); yaxis(-0.5,17.8,Ticks(f, 2.0),Arrow()); real f(real x) { return -0.03125 x^(3) + 0.75x^(2) - 5.125 x + 14.5; } draw(graph(f,0,15.225),currentpen+1); real dpt=2; real ts=0.75; transform st=scale(ts); label(rotate(90)*st*"Elevation (meters)",(-dpt,8)); label(st*"Time (seconds)",(8,-dpt)); [/asy] $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14$

Solution

Problem 10

Harold made a plum pie to take on a picnic. He was able to eat only $\frac{1}{4}$ of the pie, and he left the rest for his friends. A moose came by and ate $\frac{1}{3}$ of what Harold left behind. After that, a porcupine ate $\frac{1}{3}$ of what the moose left behind. How much of the original pie still remained after the porcupine left?

$\textbf{(A)}\ \frac{1}{12} \qquad \textbf{(B)}\ \frac{1}{6} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{1}{3} \qquad \textbf{(E)}\ \frac{5}{12}$

Solution

Problem 11

NASA’s Perseverance Rover was launched on July $30,$ $2020.$ After traveling $292{,}526{,}838$ miles, it landed on Mars in Jezero Crater about $6.5$ months later. Which of the following is closest to the Rover’s average interplanetary speed in miles per hour?

$\textbf{(A)}\ 6{,}000 \qquad \textbf{(B)}\ 12{,}000 \qquad \textbf{(C)}\ 60{,}000 \qquad \textbf{(D)}\ 120{,}000 \qquad \textbf{(E)}\ 600{,}000$

Solution

Problem 12

The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded?

[asy] // Diagram by TheMathGuyd size(6cm); draw(circle((3,3),3)); filldraw(circle((2,3),2),lightgrey); filldraw(circle((3,3),1),white); filldraw(circle((1,3),1),white); filldraw(circle((5.5,3),0.5),lightgrey); filldraw(circle((4.5,4.5),0.5),lightgrey); filldraw(circle((4.5,1.5),0.5),lightgrey); int i, j; for(i=0; i<7; i=i+1) { draw((0,i)--(6,i), dashed+grey); draw((i,0)--(i,6), dashed+grey); } [/asy]

$\textbf{(A)}\ \frac{1}{4} \qquad \textbf{(B)}\ \frac{11}{36} \qquad \textbf{(C)}\ \frac{1}{3} \qquad \textbf{(D)}\ \frac{19}{36} \qquad \textbf{(E)}\ \frac{5}{9}$


Solution

Problem 13

Along the route of a bicycle race, $7$ water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also $2$ repair stations evenly spaced between the start and finish lines. The $3$rd water station is located $2$ miles after the $1$st repair station. How long is the race in miles? [asy] // Credits given to Themathguyd‎ and Kante314 usepackage("mathptmx"); size(10cm); filldraw((11,4.5)--(171,4.5)--(171,17.5)--(11,17.5)--cycle,mediumgray*0.4 + lightgray*0.6); draw((11,11)--(171,11),linetype("2 2")+white+linewidth(1.2)); draw((0,0)--(11,0)--(11,22)--(0,22)--cycle); draw((171,0)--(182,0)--(182,22)--(171,22)--cycle); draw((31,4.5)--(31,0)); draw((51,4.5)--(51,0)); draw((151,4.5)--(151,0)); label(scale(.85)*rotate(45)*"Water 1", (23,-13.5)); label(scale(.85)*rotate(45)*"Water 2", (43,-13.5)); label(scale(.85)*rotate(45)*"Water 7", (143,-13.5)); filldraw(circle((103,-13.5),.2)); filldraw(circle((98,-13.5),.2)); filldraw(circle((93,-13.5),.2)); filldraw(circle((88,-13.5),.2)); filldraw(circle((83,-13.5),.2)); label(scale(.85)*rotate(90)*"Start", (5.5,11)); label(scale(.85)*rotate(270)*"Finish", (176.5,11)); [/asy] $\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 96$

Solution

Problem 14

Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of $5$-cent, $10$-cent, and $25$-cent stamps, with exactly $20$ of each type. What is the greatest number of stamps Nicolas can use to make exactly $$7.10$ in postage? (Note: The amount $$7.10$ corresponds to $7$ dollars and $10$ cents. One dollar is worth $100$ cents.)

$\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 54\qquad \textbf{(E)}\ 55$

Solution

Problem 15

Viswam walks half a mile to get to school each day. His route consists of $10$ city blocks of equal length and he takes $1$ minute to walk each block. Today, after walking $5$ blocks, Viswam discovers he has to make a detour, walking $3$ blocks of equal length instead of $1$ block to reach the next corner. From the time he starts his detour, at what speed, in mph, must he walk, in order to get to school at his usual time? [asy] // Diagram by TheMathGuyd size(13cm); // this is an important stickman to the left of the origin pair C=midpoint((-0.5,0.5)--(-0.6,0.05)); draw((-0.5,0.5)--(-0.6,0.05)); // Head to butt draw((-0.64,0.16)--(-0.7,0.2)--C--(-0.47,0.2)--(-0.4,0.22)); // LH-C-RH draw((-0.6,0.05)--(-0.55,-0.1)--(-0.57,-0.25)); draw((-0.6,0.05)--(-0.68,-0.12)--(-0.8,-0.20)); filldraw(circle((-0.5,0.5),0.1),white,black); int i; real d,s; // gap and side d=0.2; s=1-2*d; for(i=0; i<10; i=i+1) { //dot((i,0), red); //marks to start filldraw((i+d,d)--(i+1-d,d)--(i+1-d,1-d)--(i+d,1-d)--cycle, lightgrey, black); filldraw(conj((i+d,d))--conj((i+1-d,d))--conj((i+1-d,1-d))--conj((i+d,1-d))--cycle,lightgrey,black); } fill((5+d,-d/2)--(6-d,-d/2)--(6-d,d/2)--(5+d,d/2)--cycle,lightred); draw((0,0)--(5,0)--(5,1)--(6,1)--(6,0)--(10.1,0),deepblue+linewidth(1.25)); //Who even noticed label("School", (10,0),E, Draw()); [/asy] $\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 4.2 \qquad \textbf{(C)}\ 4.5 \qquad \textbf{(D)}\ 4.8 \qquad \textbf{(E)}\ 5$

Solution

Problem 16

The letters $\text{P}, \text{Q},$ and $\text{R}$ are entered into a $20\times20$ table according to the pattern shown below. How many $\text{P}$s, $\text{Q}$s, and $\text{R}$s will appear in the completed table? [asy] /* Made by MRENTHUSIASM, Edited by Kante314 */ usepackage("mathdots"); size(5cm); draw((0,0)--(6,0),linewidth(1.5)+mediumgray); draw((0,1)--(6,1),linewidth(1.5)+mediumgray); draw((0,2)--(6,2),linewidth(1.5)+mediumgray); draw((0,3)--(6,3),linewidth(1.5)+mediumgray); draw((0,4)--(6,4),linewidth(1.5)+mediumgray); draw((0,5)--(6,5),linewidth(1.5)+mediumgray); draw((0,0)--(0,6),linewidth(1.5)+mediumgray); draw((1,0)--(1,6),linewidth(1.5)+mediumgray); draw((2,0)--(2,6),linewidth(1.5)+mediumgray); draw((3,0)--(3,6),linewidth(1.5)+mediumgray); draw((4,0)--(4,6),linewidth(1.5)+mediumgray); draw((5,0)--(5,6),linewidth(1.5)+mediumgray); label(scale(.9)*"\textsf{P}", (.5,.5)); label(scale(.9)*"\textsf{Q}", (.5,1.5)); label(scale(.9)*"\textsf{R}", (.5,2.5)); label(scale(.9)*"\textsf{P}", (.5,3.5)); label(scale(.9)*"\textsf{Q}", (.5,4.5)); label("$\vdots$", (.5,5.6)); label(scale(.9)*"\textsf{Q}", (1.5,.5)); label(scale(.9)*"\textsf{R}", (1.5,1.5)); label(scale(.9)*"\textsf{P}", (1.5,2.5)); label(scale(.9)*"\textsf{Q}", (1.5,3.5)); label(scale(.9)*"\textsf{R}", (1.5,4.5)); label("$\vdots$", (1.5,5.6)); label(scale(.9)*"\textsf{R}", (2.5,.5)); label(scale(.9)*"\textsf{P}", (2.5,1.5)); label(scale(.9)*"\textsf{Q}", (2.5,2.5)); label(scale(.9)*"\textsf{R}", (2.5,3.5)); label(scale(.9)*"\textsf{P}", (2.5,4.5)); label("$\vdots$", (2.5,5.6)); label(scale(.9)*"\textsf{P}", (3.5,.5)); label(scale(.9)*"\textsf{Q}", (3.5,1.5)); label(scale(.9)*"\textsf{R}", (3.5,2.5)); label(scale(.9)*"\textsf{P}", (3.5,3.5)); label(scale(.9)*"\textsf{Q}", (3.5,4.5)); label("$\vdots$", (3.5,5.6)); label(scale(.9)*"\textsf{Q}", (4.5,.5)); label(scale(.9)*"\textsf{R}", (4.5,1.5)); label(scale(.9)*"\textsf{P}", (4.5,2.5)); label(scale(.9)*"\textsf{Q}", (4.5,3.5)); label(scale(.9)*"\textsf{R}", (4.5,4.5)); label("$\vdots$", (4.5,5.6)); label(scale(.9)*"$\dots$", (5.5,.5)); label(scale(.9)*"$\dots$", (5.5,1.5)); label(scale(.9)*"$\dots$", (5.5,2.5)); label(scale(.9)*"$\dots$", (5.5,3.5)); label(scale(.9)*"$\dots$", (5.5,4.5)); label(scale(.9)*"$\iddots$", (5.5,5.6)); [/asy] $\textbf{(A)}~132\text{ Ps, }134\text{ Qs, }134\text{ Rs}$

$\textbf{(B)}~133\text{ Ps, }133\text{ Qs, }134\text{ Rs}$

$\textbf{(C)}~133\text{ Ps, }134\text{ Qs, }133\text{ Rs}$

$\textbf{(D)}~134\text{ Ps, }132\text{ Qs, }134\text{ Rs}$

$\textbf{(E)}~134\text{ Ps, }133\text{ Qs, }133\text{ Rs}$

Solution

Problem 17

A regular octahedron has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of $Q$?

[asy] // Diagram by TheMathGuyd import graph; // The Solid // To save processing time, do not use three (dimensions) // Project (roughly) to two size(15cm); pair Fr, Lf, Rt, Tp, Bt, Bk; Lf=(0,0); Rt=(12,1); Fr=(7,-1); Bk=(5,2); Tp=(6,6.7); Bt=(6,-5.2); draw(Lf--Fr--Rt); draw(Lf--Tp--Rt); draw(Lf--Bt--Rt); draw(Tp--Fr--Bt); draw(Lf--Bk--Rt,dashed); draw(Tp--Bk--Bt,dashed); label(rotate(-8.13010235)*slant(0.1)*"$Q$", (4.2,1.6)); label(rotate(21.8014095)*slant(-0.2)*"$?$", (8.5,2.05)); pair g = (-8,0); // Define Gap transform real a = 8; draw(g+(-a/2,1)--g+(a/2,1), Arrow()); // Make arrow // Time for the NET pair DA,DB,DC,CD,O; DA = (4*sqrt(3),0); DB = (2*sqrt(3),6); DC = (DA+DB)/3; CD = conj(DC); O=(0,0); transform trf=shift(3g+(0,3)); path NET = O--(-2*DA)--(-2DB)--(-DB)--(2DA-DB)--DB--O--DA--(DA-DB)--O--(-DB)--(-DA)--(-DA-DB)--(-DB); draw(trf*NET); label("$7$",trf*DC); label("$Q$",trf*DC+DA-DB); label("$5$",trf*DC-DB); label("$3$",trf*DC-DA-DB); label("$6$",trf*CD); label("$4$",trf*CD-DA); label("$2$",trf*CD-DA-DB); label("$1$",trf*CD-2DA); [/asy]

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

Solution

Problem 18

Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump $5$ pads to the right or $3$ pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located $2023$ pads to the right of her starting point?

$\textbf{(A)}\ 405 \qquad \textbf{(B)}\ 407 \qquad \textbf{(C)}\ 409 \qquad \textbf{(D)}\ 411 \qquad \textbf{(E)}\ 413$

Solution

Problem 19

An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\frac23$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?

[asy] // Diagram by TheMathGuyd pair A,B,C; A=(0,1); B=(sqrt(3)/2,-1/2); C=-conj(B); fill(2B--3B--3C--2C--cycle,grey); dot(3A); dot(3B); dot(3C); dot(2A); dot(2B); dot(2C); draw(2A--2B--2C--cycle); draw(3A--3B--3C--cycle); draw(2A--3A); draw(2B--3B); draw(2C--3C); [/asy]

$\textbf{(A) } 1 : 3 \qquad \textbf{(B) } 3 : 8 \qquad \textbf{(C) } 5 : 12 \qquad \textbf{(D) } 7 : 16 \qquad \textbf{(E) } 4 : 9$

Solution

Problem 20

Two integers are inserted into the list $3,3,8,11,28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?

$\textbf{(A)}\ 56 \qquad \textbf{(B)}\ 57 \qquad \textbf{(C)}\ 58 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 61$

Solution

Problem 21

Alina writes the numbers $1, 2, \dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?

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

Solution

Problem 22

In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is $4000$. What is the first term?

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 10$

Solution

Problem 23

Each square in a $3 \times 3$ grid is randomly filled with one of the $4$ gray and white tiles shown below on the right. [asy] size(5.663333333cm); draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,gray); draw((1,0)--(1,3)--(2,3)--(2,0),gray); draw((0,1)--(3,1)--(3,2)--(0,2),gray); fill((6,.33)--(7,.33)--(7,1.33)--cycle,mediumgray); draw((6,.33)--(7,.33)--(7,1.33)--(6,1.33)--cycle,gray); fill((6,1.67)--(7,2.67)--(6,2.67)--cycle,mediumgray); draw((6,1.67)--(7,1.67)--(7,2.67)--(6,2.67)--cycle,gray); fill((7.33,.33)--(8.33,.33)--(7.33,1.33)--cycle,mediumgray); draw((7.33,.33)--(8.33,.33)--(8.33,1.33)--(7.33,1.33)--cycle,gray); fill((8.33,1.67)--(8.33,2.67)--(7.33,2.67)--cycle,mediumgray); draw((7.33,1.67)--(8.33,1.67)--(8.33,2.67)--(7.33,2.67)--cycle,gray); [/asy] What is the probability that the tiling will contain a large gray diamond in one of the smaller $2 \times 2$ grids? Below is an example of such tiling. [asy] size(2cm); fill((1,0)--(0,1)--(0,2)--(1,1)--cycle,mediumgray); fill((2,0)--(3,1)--(2,2)--(1,1)--cycle,mediumgray); fill((1,2)--(1,3)--(0,3)--cycle,mediumgray); fill((1,2)--(2,2)--(2,3)--cycle,mediumgray); fill((3,2)--(3,3)--(2,3)--cycle,mediumgray); draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,gray); draw((1,0)--(1,3)--(2,3)--(2,0),gray); draw((0,1)--(3,1)--(3,2)--(0,2),gray); [/asy]

$\textbf{(A) } \frac{1}{1024} \qquad \textbf{(B) } \frac{1}{256} \qquad \textbf{(C) } \frac{1}{64} \qquad \textbf{(D) } \frac{1}{16} \qquad \textbf{(E) } \frac{1}{4}$

Solution

Problem 24

Isosceles triangle $ABC$ has equal side lengths $AB$ and $BC$. In the figures below, segments are drawn parallel to $\overline{AC}$ so that the shaded portions of $\triangle ABC$ have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height $h$ of $\triangle ABC$?

[asy] //Diagram by TheMathGuyd size(12cm); real h = 2.5; // height real g=4; //c2c space real s = 0.65; //Xcord of Hline real adj = 0.08; //adjust line diffs pair A,B,C; B=(0,h); C=(1,0); A=-conj(C); pair PONE=(s,h*(1-s)); //Endpoint of Hline ONE pair PTWO=(s+adj,h*(1-s-adj)); //Endpoint of Hline ONE path LONE=PONE--(-conj(PONE)); //Hline ONE path LTWO=PTWO--(-conj(PTWO)); path T=A--B--C--cycle; //Triangle fill (shift(g,0)*(LTWO--B--cycle),mediumgrey); fill (LONE--A--C--cycle,mediumgrey); draw(LONE); draw(T); label("$A$",A,SW); label("$B$",B,N); label("$C$",C,SE); draw(shift(g,0)*LTWO); draw(shift(g,0)*T); label("$A$",shift(g,0)*A,SW); label("$B$",shift(g,0)*B,N); label("$C$",shift(g,0)*C,SE); draw(B--shift(g,0)*B,dashed); draw(C--shift(g,0)*A,dashed); draw((g/2,0)--(g/2,h),dashed); draw((0,h*(1-s))--B,dashed); draw((g,h*(1-s-adj))--(g,0),dashed); label("$5$", midpoint((g,h*(1-s-adj))--(g,0)),UnFill); label("$h$", midpoint((g/2,0)--(g/2,h)),UnFill); label("$11$", midpoint((0,h*(1-s))--B),UnFill); [/asy]

$\textbf{(A)}\ 14.6 \qquad \textbf{(B)}\ 14.8 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 15.2 \qquad \textbf{(E)}\ 15.4$

Solution

Problem 25

Fifteen integers $a_1, a_2, a_3, \dots, a_{15}$ are arranged in order on a number line. The integers are equally spaced and have the property that \[1 \le a_1 \le 10, \thickspace 13 \le a_2 \le 20, \thickspace \text{ and } \thickspace 241 \le a_{15}\le 250.\] What is the sum of digits of $a_{14}?$

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

Solution

See Also

2023 AMC 8 (Problems • Answer Key • Resources)
Preceded by
2022 AMC 8 		Followed by
2024 AMC 8
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 AJHSME/AMC 8 Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_AMC_8_Problems&oldid=228767"