2024 AMC 8 -
Part of AMC 8
Subcontests
(25)Roll of Tape
A roll of tape is 4 inches in diameter and is wrapped around a ring that is 2 inches in diameter. A cross section of the tape is shown in the figure below. The tape is 0.015 inches thick. If the tape is completely unrolled, approximately how long would it be? Round your answer to the nearest 100 inches.[asy]
/* AMC8 P22 2024, revised by Teacher David */
size(120);pair o = (0,0);
real r1 = 1;
real r2 = 2;filldraw(circle(o, r2), mediumgray, linewidth(1pt));
filldraw(circle(o, r1), white, linewidth(1pt));draw((-2,-2.6)--(-2,-2.4));
draw((2,-2.6)--(2,-2.4));
draw((-2,-2.5)--(2,-2.5), L=Label("4 in."));draw((-1,0)--(1,0), L=Label("2 in.", align=(0,1)), arrow=Arrows());draw((2,0)--(2,-1.3), linewidth(1pt));
[/asy]
<spanclass=′latex−bold′>(A)</span>300<spanclass=′latex−bold′>(B)</span>600<spanclass=′latex−bold′>(C)</span>1200<spanclass=′latex−bold′>(D)</span>1500<spanclass=′latex−bold′>(E)</span>1800 Complementary Counting
Four squares of side length 4,7,9, and 10 are arranged in increasing size order so that their left edges and bottom edges align. The squares alternate in color white-gray-white-gray, respectively, as shown in the figure. What is the area of the visible gray region in square units?
[asy]
size(150);
filldraw((0,0)--(10,0)--(10,10)--(0,10)--cycle,gray(0.7),linewidth(1));
filldraw((0,0)--(9,0)--(9,9)--(0,9)--cycle,white,linewidth(1));
filldraw((0,0)--(7,0)--(7,7)--(0,7)--cycle,gray(0.7),linewidth(1));
filldraw((0,0)--(4,0)--(4,4)--(0,4)--cycle,white,linewidth(1));
draw((11,0)--(11,4),linewidth(1));
draw((11,6)--(11,10),linewidth(1));
label("10",(11,5),fontsize(14pt));
draw((10.75,0)--(11.25,0),linewidth(1));
draw((10.75,10)--(11.25,10),linewidth(1));
draw((0,11)--(3,11),linewidth(1));
draw((5,11)--(9,11),linewidth(1));
draw((0,11.25)--(0,10.75),linewidth(1));
draw((9,11.25)--(9,10.75),linewidth(1));
label("9",(4,11),fontsize(14pt));
draw((-1,0)--(-1,1),linewidth(1));
draw((-1,3)--(-1,7),linewidth(1));
draw((-1.25,0)--(-0.75,0),linewidth(1));
draw((-1.25,7)--(-0.75,7),linewidth(1));
label("7",(-1,2),fontsize(14pt));
draw((0,-1)--(1,-1),linewidth(1));
draw((3,-1)--(4,-1),linewidth(1));
draw((0,-1.25)--(0,-.75),linewidth(1));
draw((4,-1.25)--(4,-.75),linewidth(1));
label("4",(2,-1),fontsize(14pt));
[/asy]
<spanclass=′latex−bold′>(A)</span> 42<spanclass=′latex−bold′>(B)</span> 45<spanclass=′latex−bold′>(C)</span> 49<spanclass=′latex−bold′>(D)</span> 50<spanclass=′latex−bold′>(E)</span> 52
Angles in Concentric Circle
Three concentric circles centered at O have radii of 1, 2, and 3. Points B and C lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle BOC, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of ∠BOC in degrees?[asy]
size(100);
import graph;draw(circle((0,0),3));
real radius = 3;
real angleStart = -54; // starting angle of the sector
real angleEnd = 54; // ending angle of the sector
label("O",(0,0),W);
pair O = (0, 0);
filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, lightgray);
filldraw(circle((0,0),2),lightgray);
filldraw(circle((0,0),1),white);
draw((1.763,2.427)--(0,0)--(1.763,-2.427));
label("B",(1.763,2.427),NE);
label("C",(1.763,-2.427),SE);
[/asy]<spanclass=′latex−bold′>(A)</span> 108<spanclass=′latex−bold′>(B)</span> 120<spanclass=′latex−bold′>(C)</span> 135<spanclass=′latex−bold′>(D)</span> 144<spanclass=′latex−bold′>(E)</span> 150
"Attacking" combo
A chess king is said to ''attack'' all squares one step away from it (basically any square right next to it in any direction), horizontally, vertically, or diagonally. For instance, a king on the center square of a 3 x 3 grid attacks all 8 other squares, as shown below. Suppose a white king and a black king are placed on different squares of 3 x 3 grid so that they do not attack each other. In how many ways can this be done?[asy]
/* AMC8 P17 2024, revised by Teacher David */
unitsize(29pt);
import math;
add(grid(3,3));pair [] a = {(0.5,0.5), (0.5, 1.5), (0.5, 2.5), (1.5, 2.5), (2.5,2.5), (2.5,1.5), (2.5,0.5), (1.5,0.5)};for (int i=0; ilabel("K", (1.5,1.5));
[/asy]<spanclass=′latex−bold′>(A)</span> 20<spanclass=′latex−bold′>(B)</span> 24<spanclass=′latex−bold′>(C)</span> 27<spanclass=′latex−bold′>(D)</span> 28<spanclass=′latex−bold′>(E)</span> 32
Graph Theory?!?!??!?2.?!!>2r
The one-way routes connecting towns A, M, C, X, Y, and Z are shown in the figure below (not drawn to scale).The distances in kilometers along each route are marked. Traveling along these routes, what is the shortest distance from A to Z in kilometers?[asy]
/* AMC8 P14 2024, by NUMANA: BUI VAN HIEU */
import graph;
unitsize(2cm);
real r=0.25;
// Define the nodes and their positions
pair[] nodes = { (0,0), (2,0), (1,1), (3,1), (4,0), (6,0) };
string[] labels = { "A", "M", "X", "Y", "C", "Z" };// Draw the nodes as circles with labels
for(int i = 0; i < nodes.length; ++i) {
draw(circle(nodes, r));
label(""+labels+"", nodes);
}
// Define the edges with their node indices and labels
int[][] edges = { {0, 1}, {0, 2}, {2, 1}, {2, 3}, {1, 3}, {1, 4}, {3, 4}, {4, 5}, {3, 5} };
string[] edgeLabels = { "8", "5", "2", "10", "6", "14", "5", "10", "17" };
pair[] edgeLabelsPos = { S, SE, SW, S, SE, S, SW, S, NE};
// Draw the edges with labels
for (int i = 0; i < edges.length; ++i) {
pair start = nodes[edges[0]];
pair end = nodes[edges[1]];
draw(start + r*dir(end-start) -- end-r*dir(end-start), Arrow);
label(""+edgeLabels+"", midpoint(start -- end), edgeLabelsPos);
}
// Draw the curved edge with label
draw(nodes[1]+r * dir(-45)..controls (3, -0.75) and (5, -0.75)..nodes[5]+r * dir(-135), Arrow);
label("25", midpoint(nodes[1]..controls (3, -0.75) and (5, -0.75)..nodes[5]), 2S);
[/asy]<spanclass=′latex−bold′>(A)</span> 28<spanclass=′latex−bold′>(B)</span> 29<spanclass=′latex−bold′>(C)</span> 30<spanclass=′latex−bold′>(D)</span> 31<spanclass=′latex−bold′>(E)</span> 32
Length comparison
Sergei skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the paths labeled P, Q, R, and S. What is the sorted order of the four paths from shortest to longest?
[img width=50]https://wiki-images.artofproblemsolving.com//9/94/2024_AMC_8_Problem_6.png<spanclass=′latex−bold′>(A)</span>P, Q, R, S<spanclass=′latex−bold′>(B)</span>P, R, S, Q<spanclass=′latex−bold′>(C)</span>Q, S, P, R<spanclass=′latex−bold′>(D)</span>R, P, S, Q<spanclass=′latex−bold′>(E)</span>R, S, P, Q CUBES LOOK cOoL
Any three vertices of the cube PQRSTUVW, shown in the figure below, can be connected to form a triangle. (For example, vertices P,Q, and R can be connected to form △PQR.) How many of these triangles are equilateral and contain P as a vertex?<spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>2<spanclass=′latex−bold′>(D)</span>3<spanclass=′latex−bold′>(E)</span>6
[asy]
unitsize(4);
pair P,Q,R,S,T,U,V,W;
P=(0,30); Q=(30,30); R=(40,40); S=(10,40); T=(10,10); U=(40,10); V=(30,0); W=(0,0);
draw(W--V); draw(V--Q); draw(Q--P); draw(P--W); draw(T--U); draw(U--R); draw(R--S); draw(S--T); draw(W--T); draw(P--S); draw(V--U); draw(Q--R);
dot(P);
dot(Q);
dot(R);
dot(S);
dot(T);
dot(U);
dot(V);
dot(W);
label("P",P,NW);
label("Q",Q,NW);
label("R",R,NE);
label("S",S,N);
label("T",T,NE);
label("U",U,NE);
label("V",V,SE);
label("W",W,SW);
[/asy] Stained Glass Artwork
Jean made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is 8 feet high and the other peak is 12 feet high. Each peak forms a 90∘ angle, and the straight sides of the mountains form 45∘ with the ground. The artwork has an area of 183 square feet. The sides of the mountains meet at an intersection point near the center of the artwork, h feet above the ground. What is the value of h?[asy]
unitsize(.3cm);
filldraw((0,0)--(8,8)--(11,5)--(18,12)--(30,0)--cycle,gray(0.7),linewidth(1));
draw((-1,0)--(-1,8),linewidth(.75));
draw((-1.4,0)--(-.6,0),linewidth(.75));
draw((-1.4,8)--(-.6,8),linewidth(.75));
label("8",(-1,4),W);
label("12",(31,6),E);
draw((-1,8)--(8,8),dashed);
draw((31,0)--(31,12),linewidth(.75));
draw((30.6,0)--(31.4,0),linewidth(.75));
draw((30.6,12)--(31.4,12),linewidth(.75));
draw((31,12)--(18,12),dashed);
label("45∘",(.75,0),NE,fontsize(10pt));
label("45∘",(29.25,0),NW,fontsize(10pt));
draw((8,8)--(7.5,7.5)--(8,7)--(8.5,7.5)--cycle);
draw((18,12)--(17.5,11.5)--(18,11)--(18.5,11.5)--cycle);
draw((11,5)--(11,0),dashed);
label("h",(11,2.5),E);
[/asy]<spanclass=′latex−bold′>(A)</span> 4<spanclass=′latex−bold′>(B)</span> 5<spanclass=′latex−bold′>(C)</span> 42<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 52Credits to Countmath1 for the diagram. Classic Triangle Area problem
The coordinates of △ABC are A(5,7), B(11,7), C(3,y), with y>7. The area of △ABC is 12. What is the value of y?[asy]
size(10cm);
draw((5,7)--(11,7)--(3,11)--cycle);
label("A(5,7)", (5,7),S);
label("B(11,7)", (11,7),S);
label("C(3,y)", (3,11),W);
[/asy]<spanclass=′latex−bold′>(A)</span>8<spanclass=′latex−bold′>(B)</span>9<spanclass=′latex−bold′>(C)</span>10<spanclass=′latex−bold′>(D)</span>11<spanclass=′latex−bold′>(E)</span>12 Linecense
Rodrigo has a very large sheet of graph paper. First he draws a line segment connecting point (0,4) to point (2,0) and colors the 4 cells whose interiors intersect the segment, as shown below. Next Rodrigo draws a line segment connecting point (2000,3000) to point (5000,8000). How many cells will he color this time?[asy]
filldraw((0,4)--(1,4)--(1,3)--(0,3)--cycle, gray(.75), gray(.5)+linewidth(1));
filldraw((0,3)--(1,3)--(1,2)--(0,2)--cycle, gray(.75), gray(.5)+linewidth(1));
filldraw((1,2)--(2,2)--(2,1)--(1,1)--cycle, gray(.75), gray(.5)+linewidth(1));
filldraw((1,1)--(2,1)--(2,0)--(1,0)--cycle, gray(.75), gray(.5)+linewidth(1));draw((-1,5)--(-1,-1),gray(.9));
draw((0,5)--(0,-1),gray(.9));
draw((1,5)--(1,-1),gray(.9));
draw((2,5)--(2,-1),gray(.9));
draw((3,5)--(3,-1),gray(.9));
draw((4,5)--(4,-1),gray(.9));
draw((5,5)--(5,-1),gray(.9));draw((-1,5)--(5, 5),gray(.9));
draw((-1,4)--(5,4),gray(.9));
draw((-1,3)--(5,3),gray(.9));
draw((-1,2)--(5,2),gray(.9));
draw((-1,1)--(5,1),gray(.9));
draw((-1,0)--(5,0),gray(.9));
draw((-1,-1)--(5,-1),gray(.9));
dot((0,4));
label("(0,4)",(0,4),NW);dot((2,0));
label("(2,0)",(2,0),SE);draw((0,4)--(2,0));draw((-1,0) -- (5,0), arrow=Arrow);
draw((0,-1) -- (0,5), arrow=Arrow);[/asy]<spanclass=′latex−bold′>(A)</span>6000<spanclass=′latex−bold′>(B)</span>6500<spanclass=′latex−bold′>(C)</span>7000<spanclass=′latex−bold′>(D)</span>7500<spanclass=′latex−bold′>(E)</span>8000