Difference between revisions of "2020 AMC 8 Problems"
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Akash's birthday cake is in the form of a <math>4 \times 4 \times 4</math> 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 <math>64</math> smaller cubes, each measuring <math>1 \times 1 \times 1</math> inch, as shown below. How many small pieces will have icing on exactly two sides? | Akash's birthday cake is in the form of a <math>4 \times 4 \times 4</math> 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 <math>64</math> smaller cubes, each measuring <math>1 \times 1 \times 1</math> inch, as shown below. How many 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> | ||
<math>\textbf{(A) }\text{12} \qquad \textbf{(B) }\text{16} \qquad \textbf{(C) }\text{18} \qquad \textbf{(D) }\text{20}\qquad \textbf{(E) }\text{24}</math> | <math>\textbf{(A) }\text{12} \qquad \textbf{(B) }\text{16} \qquad \textbf{(C) }\text{18} \qquad \textbf{(D) }\text{20}\qquad \textbf{(E) }\text{24}</math> | ||
[[2020 AMC 8 Problems/Problem 9|Solution]] | [[2020 AMC 8 Problems/Problem 9|Solution]] |
Revision as of 23:05, 17 November 2020
Contents
[hide]Problem 1
Luka is making lemonade to sell at a school fundraiser. His recipe requires times as much water as sugar and twice as much sugar as lemon juice. He uses cups of lemon juice. How many cups of water does he need?
Problem 2
Four friends do yardwork for their neighbors over the weekend, earning and respectively. They decide to split their earnings equally among themselves. In total how much will the friend who earned give to the others?
Problem 3
Carrie has a rectangular garden that measures feet by feet. She plants the entire garden with strawberry plants. Carrie is able to plant strawberry plants per square foot, and she harvests an average of strawberries per plant. How many strawberries can she expect to harvest?
Problem 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?
Diagram by sircalcsalot
Problem 5
Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of cups. What percent of the total capacity of the pitcher did each cup receive?
Problem 6
Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?
Problem 7
How many integers between and have four distinct digits arranged in increasing order? (For example, is one integer.)
Problem 8
Ricardo has coins, some of which are pennies (-cent coins) and the rest of which are nickels (-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?
Problem 9
Akash's birthday cake is in the form of a 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 smaller cubes, each measuring inch, as shown below. How many small pieces will have icing on exactly two sides?