Difference between revisions of "2023 AMC 8 Problems"
MRENTHUSIASM (talk | contribs) (Undo revision 228745 by Ben.iriarte (talk)) (Tag: Undo) |
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{{AMC8 Problems|year=2023|}} | {{AMC8 Problems|year=2023|}} | ||
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==Problem 1== | ==Problem 1== | ||
Line 13: | Line 10: | ||
==Problem 2== | ==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> | ||
[[2023 AMC 8 Problems/Problem 2|Solution]] | [[2023 AMC 8 Problems/Problem 2|Solution]] | ||
Line 23: | 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> | + | 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{( | + | <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 48: | 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? | + | |
+ | <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 57: | 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 81: | 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 100: | Line 376: | ||
==Problem 11== | ==Problem 11== | ||
− | NASA’s Perseverance Rover was launched on July <math>30,</math> <math>2020.</math> After traveling <math>292,526,838</math> miles, it landed on Mars in Jezero Crater about <math>6.5</math> months later. Which of the following is closest to the Rover’s average interplanetary speed in miles per hour? | + | NASA’s Perseverance Rover was launched on July <math>30,</math> <math>2020.</math> After traveling <math>292{,}526{,}838</math> miles, it landed on Mars in Jezero Crater about <math>6.5</math> months later. Which of the following is closest to the Rover’s average interplanetary speed in miles per hour? |
− | <math>\textbf{(A)}\ 6,000 \qquad \textbf{(B)}\ 12,000 \qquad \textbf{(C)}\ 60,000 \qquad \textbf{(D)}\ 120,000 \qquad \textbf{(E)}\ 600,000</math> | + | <math>\textbf{(A)}\ 6{,}000 \qquad \textbf{(B)}\ 12{,}000 \qquad \textbf{(C)}\ 60{,}000 \qquad \textbf{(D)}\ 120{,}000 \qquad \textbf{(E)}\ 600{,}000</math> |
[[2023 AMC 8 Problems/Problem 11|Solution]] | [[2023 AMC 8 Problems/Problem 11|Solution]] | ||
Line 109: | Line 385: | ||
The figure below shows a large white circle with a number of smaller white and shaded circles in its | 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? | 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> | ||
<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]] | ||
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==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 | + | 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 126: | 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> | ||
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==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> | ||
+ | // 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> | ||
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 4.2 \qquad \textbf{(C)}\ 4.5 \qquad \textbf{(D)}\ 4.8 \qquad \textbf{(E)}\ 5</math> | <math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 4.2 \qquad \textbf{(C)}\ 4.5 \qquad \textbf{(D)}\ 4.8 \qquad \textbf{(E)}\ 5</math> | ||
[[2023 AMC 8 Problems/Problem 15|Solution]] | [[2023 AMC 8 Problems/Problem 15|Solution]] | ||
+ | |||
+ | ==Problem 16== | ||
+ | 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)); | ||
+ | |||
+ | 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> | ||
+ | <math>\textbf{(A)}~132\text{ Ps, }134\text{ Qs, }134\text{ Rs}</math> | ||
+ | |||
+ | <math>\textbf{(B)}~133\text{ Ps, }133\text{ Qs, }134\text{ Rs}</math> | ||
+ | |||
+ | <math>\textbf{(C)}~133\text{ Ps, }134\text{ Qs, }133\text{ Rs}</math> | ||
+ | |||
+ | <math>\textbf{(D)}~134\text{ Ps, }132\text{ Qs, }134\text{ Rs}</math> | ||
+ | |||
+ | <math>\textbf{(E)}~134\text{ Ps, }133\text{ Qs, }133\text{ Rs}</math> | ||
+ | |||
+ | [[2023 AMC 8 Problems/Problem 16|Solution]] | ||
==Problem 17== | ==Problem 17== | ||
− | A < | + | 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> | ||
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==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>\ | + | <asy> |
− | the inner triangle? | + | // 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> | ||
− | <math>\textbf{(A)} | + | <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]] | ||
− | |||
− | |||
− | |||
− | |||
==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 179: | 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 | + | 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 186: | 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 193: | Line 670: | ||
[[2023 AMC 8 Problems/Problem 22|Solution]] | [[2023 AMC 8 Problems/Problem 22|Solution]] | ||
− | == | + | ==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. | |
+ | <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 <math>2 \times 2</math> 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> | ||
<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 208: | 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 | + | 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 217: | 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: • AoPS Resources • PDF | ||
Instructions
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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 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See Also
Problem 1
What is the value of ?
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?
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 where temperature is measured in degrees Fahrenheit and the wind speed is measured in miles per hour (mph). Suppose the air temperature is and the wind speed is mph. Which of the following is closest to the approximate wind chill?
Problem 4
The numbers from to 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 How many of these four numbers are prime?
Problem 5
A lake contains trout, along with a variety of other fish. When a marine biologist catches and releases a sample of fish from the lake, 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?
Problem 6
The digits and are placed in the expression below, one digit per box. What is the maximum possible value of the expression?
Problem 7
A rectangle, with sides parallel to the -axis and -axis, has opposite vertices located at and . A line is drawn through points and . Another line is drawn through points and . How many points on the rectangle lie on at least one of the two lines?
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 and represent a win or loss, respectively. For example, Lola won five matches and lost the fourth match. What was Tiyo’s win-loss record?
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 and meters?
Problem 10
Harold made a plum pie to take on a picnic. He was able to eat only of the pie, and he left the rest for his friends. A moose came by and ate of what Harold left behind. After that, a porcupine ate of what the moose left behind. How much of the original pie still remained after the porcupine left?
Problem 11
NASA’s Perseverance Rover was launched on July After traveling miles, it landed on Mars in Jezero Crater about months later. Which of the following is closest to the Rover’s average interplanetary speed in miles per hour?
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?
Problem 13
Along the route of a bicycle race, water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also repair stations evenly spaced between the start and finish lines. The rd water station is located miles after the st repair station. How long is the race in miles?
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 -cent, -cent, and -cent stamps, with exactly of each type. What is the greatest number of stamps Nicolas can use to make exactly in postage? (Note: The amount corresponds to dollars and cents. One dollar is worth cents.)
Problem 15
Viswam walks half a mile to get to school each day. His route consists of city blocks of equal length and he takes minute to walk each block. Today, after walking blocks, Viswam discovers he has to make a detour, walking blocks of equal length instead of 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?
Problem 16
The letters and are entered into a table according to the pattern shown below. How many s, s, and s will appear in the completed table?
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 ?
Problem 18
Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump pads to the right or pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located pads to the right of her starting point?
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 the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?
Problem 20
Two integers are inserted into the list to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?
Problem 21
Alina writes the numbers on separate cards, one number per card. She wishes to divide the cards into groups of cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?
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 . What is the first term?
Problem 23
Each square in a grid is randomly filled with one of the gray and white tiles shown below on the right. What is the probability that the tiling will contain a large gray diamond in one of the smaller grids? Below is an example of such tiling.
Problem 24
Isosceles triangle has equal side lengths and . In the figures below, segments are drawn parallel to so that the shaded portions of have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height of ?
Problem 25
Fifteen integers are arranged in order on a number line. The integers are equally spaced and have the property that What is the sum of digits of
See Also
2023 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by 2022 AMC 8 |
Followed by 2024 AMC 8 | |
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All AJHSME/AMC 8 Problems and Solutions |