Found problems: 14
2020 AMC 8 -, 8
Ricardo has $2020$ coins, some of which are pennies ($1$-cent coins) and the rest of which are nickels ($5$-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least possible amounts of money that Ricardo can have?
$\textbf{(A) }8062 \qquad \textbf{(B) }8068 \qquad \textbf{(C) }8072 \qquad \textbf{(D) }8076 \qquad \textbf{(E) }8082$
2020 AMC 8 -, 10
Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }18 \qquad \textbf{(E) }24$
2020 AMC 8 -, 22
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.
[asy] size(300); defaultpen(linewidth(0.8)+fontsize(13)); real r = 0.05; draw((0.9,0)--(3.5,0),EndArrow(size=7)); filldraw((4,2.5)--(7,2.5)--(7,-2.5)--(4,-2.5)--cycle,gray(0.65)); fill(circle((5.5,1.25),0.8),white); fill(circle((5.5,1.25),0.5),gray(0.65)); fill((4.3,-r)--(6.7,-r)--(6.7,-1-r)--(4.3,-1-r)--cycle,white); fill((4.3,-1.25+r)--(6.7,-1.25+r)--(6.7,-2.25+r)--(4.3,-2.25+r)--cycle,white); fill((4.6,-0.25-r)--(6.4,-0.25-r)--(6.4,-0.75-r)--(4.6,-0.75-r)--cycle,gray(0.65)); fill((4.6,-1.5+r)--(6.4,-1.5+r)--(6.4,-2+r)--(4.6,-2+r)--cycle,gray(0.65)); label("$N$",(0.45,0)); draw((7.5,1.25)--(11.25,1.25),EndArrow(size=7)); draw((7.5,-1.25)--(11.25,-1.25),EndArrow(size=7)); label("if $N$ is even",(9.25,1.25),N); label("if $N$ is odd",(9.25,-1.25),N); label("$\frac N2$",(12,1.25)); label("$3N+1$",(12.6,-1.25)); [/asy]
For example, starting with an input of $N = 7$, the machine will output $3 \cdot 7 + 1 = 22$. Then if the output is repeatedly inserted into the machine five more times, the final output is $26$. $$ 7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26$$ When the same 6-step process is applied to a different starting value of $N$, the final output is $1$. What is the sum of all such integers $N$? $$ N \to \_\_ \to \_\_ \to \_\_ \to \_\_ \to \_\_ \to 1$$
$\textbf{(A)}\ 73 \qquad \textbf{(B)}\ 74 \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ 82 \qquad \textbf{(E)}\ 83$
2020 AMC 8 -, 4
Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?
[asy]
// diagram by SirCalcsALot
size(250);
real side1 = 1.5;
real side2 = 4.0;
real side3 = 6.5;
real pos = 2.5;
pair s1 = (-10,-2.19);
pair s2 = (15,2.19);
pen grey1 = rgb(100/256, 100/256, 100/256);
pen grey2 = rgb(183/256, 183/256, 183/256);
fill(circle(origin + s1, 1), grey1);
for (int i = 0; i < 6; ++i) {
draw(side1*dir(60*i)+s1--side1*dir(60*i-60)+s1,linewidth(1.25));
}
fill(circle(origin, 1), grey1);
for (int i = 0; i < 6; ++i) {
fill(circle(pos*dir(60*i),1), grey2);
draw(side2*dir(60*i)--side2*dir(60*i-60),linewidth(1.25));
}
fill(circle(origin+s2, 1), grey1);
for (int i = 0; i < 6; ++i) {
fill(circle(pos*dir(60*i)+s2,1), grey2);
fill(circle(2*pos*dir(60*i)+s2,1), grey1);
fill(circle(sqrt(3)*pos*dir(60*i+30)+s2,1), grey1);
draw(side3*dir(60*i)+s2--side3*dir(60*i-60)+s2,linewidth(1.25));
}
[/asy]
$\textbf{(A)}\ 35 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 39 \qquad \textbf{(D)}\ 43 \qquad \textbf{(E)}\ 49$
2020 AMC 8 -, 1
Luka is making lemonade to sell at a school fundraiser. His recipe requires $4$ times as much water as sugar and twice as much sugar as lemon juice. He uses $3$ cups of lemon juice. How many cups of water does he need?
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$
2020 AMC 8 -, 23
Five different awards are to be given to three students. Each student will receive at least one award. In how many ways can the awards be distributed?
$\textbf{(A)}\ 120 \qquad \textbf{(B)}\ 150 \qquad \textbf{(C)}\ 180 \qquad \textbf{(D)}\ 210 \qquad \textbf{(E)}\ 240$
2020 AMC 8 -, 21
A game board consists of $64$ squares that alternate in color between black and white. The figure below shows square $P$ in the bottom and square $Q$ in the top row. A marker is placed at $P$. A [i]step[/i] consists of moving the marker onto one of the adjoining white squares in the row above. How many $7$-step paths are there from $P$ to $Q$? (The figure shows a sample path.)
[asy]//diagram by SirCalcsALot
size(200); int[] x = {6, 5, 4, 5, 6, 5, 6}; int[] y = {1, 2, 3, 4, 5, 6, 7}; int N = 7; for (int i = 0; i < 8; ++i) { for (int j = 0; j < 8; ++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)); if ((i+j) % 2 == 0) { filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,black); } } } for (int i = 0; i < N; ++i) { draw(circle((x[i],y[i])+(0.5,0.5),0.35)); } label("$P$", (5.5, 0.5)); label("$Q$", (6.5, 7.5)); [/asy]
$\textbf{(A)}\ 28 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 35$
2020 AMC 8 -, 12
For a positive integer $n,$ the factorial notation $n!$ represents the product of the integers from $n$ to $1.$ (For example, $6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1.$) What value of $N$ satisfies the following equation?
$$5! \cdot 9! = 12 \cdot N!$$
$\textbf{(A) }10 \qquad \textbf{(B) }11 \qquad \textbf{(C) }12 \qquad \textbf{(D) }13 \qquad \textbf{(E) }14$
2020 AMC 8 -, 16
Each of the points $A$, $B$, $C$, $D$, $E$, and $F$ in the figure below represent a different digit from 1 to 6. Each of the five lines shown passes through some of these points. The digits along the line each are added to produce 5 sums, one for each line. The total of the sums is $47$. What is the digit represented by $B$?
[asy]
size(200);
dotfactor = 10;
pair p1 = (-28,0);
pair p2 = (-111,213);
draw(p1--p2,linewidth(1));
pair p3 = (-160,0);
pair p4 = (-244,213);
draw(p3--p4,linewidth(1));
pair p5 = (-316,0);
pair p6 = (-67,213);
draw(p5--p6,linewidth(1));
pair p7 = (0, 68);
pair p8 = (-350,10);
draw(p7--p8,linewidth(1));
pair p9 = (0, 150);
pair p10 = (-350, 62);
draw(p9--p10,linewidth(1));
pair A = intersectionpoint(p1--p2, p5--p6);
dot("$A$", A, 2*W);
pair B = intersectionpoint(p5--p6, p3--p4);
dot("$B$", B, 2*WNW);
pair C = intersectionpoint(p7--p8, p5--p6);
dot("$C$", C, 1.5*NW);
pair D = intersectionpoint(p3--p4, p7--p8);
dot("$D$", D, 2*NNE);
pair EE = intersectionpoint(p1--p2, p7--p8);
dot("$E$", EE, 2*NNE);
pair F = intersectionpoint(p1--p2, p9--p10);
dot("$F$", F, 2*NNE);
[/asy]
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
2020 AMC 8 -, 24
A large square region is paved with $n^2$ gray square tiles, each measuring $s$ inches on a side. A border $d$ inches wide surrounds each tile. The figure below shows the case for $n = 3$. When $n = 24$, the $576$ gray tiles cover $64\%$ of the area of the large square region. What is the ratio $\frac{d}{s}$ for this larger value of $n$?
[asy]
draw((0,0)--(13,0)--(13,13)--(0,13)--cycle);
filldraw((1,1)--(4,1)--(4,4)--(1,4)--cycle, mediumgray);
filldraw((1,5)--(4,5)--(4,8)--(1,8)--cycle, mediumgray);
filldraw((1,9)--(4,9)--(4,12)--(1,12)--cycle, mediumgray);
filldraw((5,1)--(8,1)--(8,4)--(5,4)--cycle, mediumgray);
filldraw((5,5)--(8,5)--(8,8)--(5,8)--cycle, mediumgray);
filldraw((5,9)--(8,9)--(8,12)--(5,12)--cycle, mediumgray);
filldraw((9,1)--(12,1)--(12,4)--(9,4)--cycle, mediumgray);
filldraw((9,5)--(12,5)--(12,8)--(9,8)--cycle, mediumgray);
filldraw((9,9)--(12,9)--(12,12)--(9,12)--cycle, mediumgray);
[/asy]
$\textbf{(A) }\frac6{25} \qquad \textbf{(B) }\frac14 \qquad \textbf{(C) }\frac9{25} \qquad \textbf{(D) }\frac7{16} \qquad \textbf{(E) }\frac9{16}$
2020 AMC 8 -, 2
Four friends do yardwork for their neighbors over the weekend, earning $\$15$, $\$20$, $\$25$, and $\$40$, respectively. They decide to split their earnings equally among themselves. In total how much will the friend who earned $\$40$ give to the others?
$\textbf{(A)}\ \$5 \qquad \textbf{(B)}\ \$10 \qquad \textbf{(C)}\ \$15\qquad \textbf{(D)}\ \$20 \qquad \textbf{(E)}\ \$25$
2020 AMC 8 -, 9
Akash's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides?
[asy]
/*
Created by SirCalcsALot and sonone
Code modfied from https://artofproblemsolving.com/community/c3114h2152994_the_old__aops_logo_with_asymptote
*/
import three;
currentprojection=orthographic(1.75,7,2);
//++++ edit colors, names are self-explainatory ++++
//pen top=rgb(27/255, 135/255, 212/255);
//pen right=rgb(254/255,245/255,182/255);
//pen left=rgb(153/255,200/255,99/255);
pen top = rgb(170/255, 170/255, 170/255);
pen left = rgb(81/255, 81/255, 81/255);
pen right = rgb(165/255, 165/255, 165/255);
pen edges=black;
int max_side = 4;
//+++++++++++++++++++++++++++++++++++++++
path3 leftface=(1,0,0)--(1,1,0)--(1,1,1)--(1,0,1)--cycle;
path3 rightface=(0,1,0)--(1,1,0)--(1,1,1)--(0,1,1)--cycle;
path3 topface=(0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle;
for(int i=0; i<max_side; ++i){
for(int j=0; j<max_side; ++j){
draw(shift(i,j,-1)*surface(topface),top);
draw(shift(i,j,-1)*topface,edges);
draw(shift(i,-1,j)*surface(rightface),right);
draw(shift(i,-1,j)*rightface,edges);
draw(shift(-1,j,i)*surface(leftface),left);
draw(shift(-1,j,i)*leftface,edges);
}
}
picture CUBE;
draw(CUBE,surface(leftface),left,nolight);
draw(CUBE,surface(rightface),right,nolight);
draw(CUBE,surface(topface),top,nolight);
draw(CUBE,topface,edges);
draw(CUBE,leftface,edges);
draw(CUBE,rightface,edges);
// begin made by SirCalcsALot
int[][] heights = {{4,4,4,4},{4,4,4,4},{4,4,4,4},{4,4,4,4}};
for (int i = 0; i < max_side; ++i) {
for (int j = 0; j < max_side; ++j) {
for (int k = 0; k < min(heights[i][j], max_side); ++k) {
add(shift(i,j,k)*CUBE);
}
}
}
[/asy]
$\textbf{(A)}\ 12\qquad~~\textbf{(B)}\ 16\qquad~~\textbf{(C)}\ 18\qquad~~\textbf{(D)}\ 20\qquad~~\textbf{(E)}\ 24$
2020 AMC 8 -, 19
A number is called [i]flippy[/i] if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15$?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$
2020 AMC 8 -, 3
Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she expect to harvest?
$\textbf{(A)}\ 560 \qquad \textbf{(B)}\ 960 \qquad \textbf{(C)}\ 1120 \qquad \textbf{(D)}\ 1920 \qquad \textbf{(E)}\ 3840$