2021 AMC 10 Fall
Part of AMC 10
Subcontests
(25)A dj geo config?
A square with side length 3 is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length 2 has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?[asy]
//diagram by kante314
draw((0,0)--(8,0)--(4,8)--cycle, linewidth(1.5));
draw((2,0)--(2,4)--(6,4)--(6,0)--cycle, linewidth(1.5));
draw((3,4)--(3,6)--(5,6)--(5,4)--cycle, linewidth(1.5));
[/asy](<spanclass=′latex−bold′>A</span>)1941(<spanclass=′latex−bold′>B</span>)2041(<spanclass=′latex−bold′>C</span>)2143(<spanclass=′latex−bold′>D</span>)2221(<spanclass=′latex−bold′>E</span>)2343 im bad at asymptote
As shown in the figure below, point E lies on the opposite half-plane determined by line CD from point A so that ∠CDE=110∘. Point F lies on AD so that DE=DF, and ABCD is a square. What is the degree measure of ∠AFE?
[asy]
size(6cm);
pair A = (0,10);
label("A", A, N);
pair B = (0,0);
label("B", B, S);
pair C = (10,0);
label("C", C, S);
pair D = (10,10);
label("D", D, SW);
pair EE = (15,11.8);
label("E", EE, N);
pair F = (3,10);
label("F", F, N);
filldraw(D--arc(D,2.5,270,380)--cycle,lightgray);
dot(A^^B^^C^^D^^EE^^F);
draw(A--B--C--D--cycle);
draw(D--EE--F--cycle);
label("110∘", (15,9), SW);
[/asy]
<spanclass=′latex−bold′>(A)</span>160<spanclass=′latex−bold′>(B)</span>164<spanclass=′latex−bold′>(C)</span>166<spanclass=′latex−bold′>(D)</span>170<spanclass=′latex−bold′>(E)</span>174 Complementary Counting
What is the area of the shaded figure shown below?[asy]
size(200);
defaultpen(linewidth(0.4)+fontsize(12));
pen s = linewidth(0.8)+fontsize(8);pair O,X,Y;
O = origin;
X = (6,0);
Y = (0,5);
fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2));
for (int i=1; i<7; ++i)
{
draw((i,0)--(i,5), gray+dashed);
label(""+string(i)+"", (i,0), 2*S);
if (i<6)
{
draw((0,i)--(6,i), gray+dashed);
label(""+string(i)+"", (0,i), 2*W);
}
}
label("0", O, 2*SW);
draw(O--X+(0.15,0), EndArrow);
draw(O--Y+(0,0.15), EndArrow);
draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5);
[/asy] Pythagorean or Similar Triag
In square ABCD, points P and Q lie on AD and AB, respectively. Segments BP and CQ intersect at right angles at R, with BR=6 and PR=7. What is the area of the square?[asy]
size(170);
defaultpen(linewidth(0.6));
real r = 3.5;
pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0),
R = intersectionpoint(B--P,C--Q);
draw(A--B--C--D--A^^B--P^^C--Q^^rightanglemark(P,R,C,7));
dot("A",A,S);
dot("B",B,S);
dot("C",C,N);
dot("D",D,N);
dot("Q",Q,S);
dot("P",P,W);
dot("R",R,1.3*S);
label("7",(P+R)/2,NE);
label("6",(R+B)/2,NE);
[/asy]
<spanclass=′latex−bold′>(A)</span>85<spanclass=′latex−bold′>(B)</span>93<spanclass=′latex−bold′>(C)</span>100<spanclass=′latex−bold′>(D)</span>117<spanclass=′latex−bold′>(E)</span>125 A weird solar panel
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon ABCDEF, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of the pillars at A, B, and C are 12,9, and 10 meters, respectively. What is the height, in meters, of the pillar at E?<spanclass=′latex−bold′>(A)</span>9<spanclass=′latex−bold′>(B)</span>63<spanclass=′latex−bold′>(C)</span>83<spanclass=′latex−bold′>(D)</span>17<spanclass=′latex−bold′>(E)</span>123 Please no silly
What is the value of 1234+2341+3412+4123? <spanclass=′latex−bold′>(A)</span>10,000<spanclass=′latex−bold′>(B)</span>10,010<spanclass=′latex−bold′>(C)</span>10,110<spanclass=′latex−bold′>(D)</span>11,000<spanclass=′latex−bold′>(E)</span>11,110 Des Moines
A farmer's rectangular field is partitioned into 2 by 2 grid of 4 rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
[asy]
draw((0,0)--(100,0)--(100,50)--(0,50)--cycle);
draw((50,0)--(50,50));
draw((0,25)--(100,25));
[/asy]
<spanclass=′latex−bold′>(A)</span> 12<spanclass=′latex−bold′>(B)</span> 64<spanclass=′latex−bold′>(C)</span> 84<spanclass=′latex−bold′>(D)</span> 90<spanclass=′latex−bold′>(E)</span> 144 Rotating Paper Problem
Three identical square sheets of paper each with side length 6 are stacked on top of each other. The middle sheet is rotated clockwise 30∘ about its center and the top sheet is rotated clockwise 60∘ about its center, resulting in the 24-sided polygon shown in the figure below. The area of this polygon can be expressed in the form a−bc, where a, b, and c are positive integers, and c is not divisible by the square of any prime. What is a+b+c?[asy]
size(160);
defaultpen(linewidth(1.1));
path square = (1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle;
filldraw(square,white);
filldraw(rotate(30)*square,white);
filldraw(rotate(60)*square,white);
dot((0,0),linewidth(7));
[/asy]<spanclass=′latex−bold′>(A)</span>75<spanclass=′latex−bold′>(B)</span>93<spanclass=′latex−bold′>(C)</span>96<spanclass=′latex−bold′>(D)</span>129<spanclass=′latex−bold′>(E)</span>147 POV: You're a student
A school has 100 students and 5 teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are 50,20,20,5, and 5. Let t be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let s be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is t−s?<spanclass=′latex−bold′>(A)</span> −18.5<spanclass=′latex−bold′>(B)</span> −13.5<spanclass=′latex−bold′>(C)</span> 0<spanclass=′latex−bold′>(D)</span> 13.5<spanclass=′latex−bold′>(E)</span> 18.5 2013 AMC 10A #25 on steroids
Regular polygons with 5,6,7, and 8 sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect? <spanclass=′latex−bold′>(A)</span> 52<spanclass=′latex−bold′>(B)</span> 56<spanclass=′latex−bold′>(C)</span> 60<spanclass=′latex−bold′>(D)</span> 64<spanclass=′latex−bold′>(E)</span> 68
#25 geo :cries:
A rectangle with side lengths 1 and 3, a square with side length 1, and a rectangle R are inscribed inside a larger square as shown. The sum of all possible values for the area of R can be written in the form nm, where m and n are relatively prime positive integers. What is m+n?[asy]
size(8cm);
draw((0,0)--(10,0));
draw((0,0)--(0,10));
draw((10,0)--(10,10));
draw((0,10)--(10,10));
draw((1,6)--(0,9));
draw((0,9)--(3,10));
draw((3,10)--(4,7));
draw((4,7)--(1,6));
draw((0,3)--(1,6));
draw((1,6)--(10,3));
draw((10,3)--(9,0));
draw((9,0)--(0,3));
draw((6,13/3)--(10,22/3));
draw((10,22/3)--(8,10));
draw((8,10)--(4,7));
draw((4,7)--(6,13/3));
label("3",(9/2,3/2),N);
label("3",(11/2,9/2),S);
label("1",(1/2,9/2),E);
label("1",(19/2,3/2),W);
label("1",(1/2,15/2),E);
label("1",(3/2,19/2),S);
label("1",(5/2,13/2),N);
label("1",(7/2,17/2),W);
label("R",(7,43/6),W);
[/asy](<spanclass=′latex−bold′>A</span>)14(<spanclass=′latex−bold′>B</span>)23(<spanclass=′latex−bold′>C</span>)46(<spanclass=′latex−bold′>D</span>)59(<spanclass=′latex−bold′>E</span>)67