Found problems: 147
2016 AMC 12/AHSME, 8
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece?
$\textbf{(A)}\ 14.0\qquad\textbf{(B)}\ 16.0\qquad\textbf{(C)}\ 20.0\qquad\textbf{(D)}\ 33.3\qquad\textbf{(E)}\ 55.6$
2020 AMC 10, 19
In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. What is the digit $A$?
$\textbf{(A) } 2 \qquad\textbf{(B) } 3 \qquad\textbf{(C) } 4 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } 7$
2016 AMC 12/AHSME, 14
The sum of an infinite geometric series is a positive number $S$, and the second term in the series is $1$. What is the smallest possible value of $S?$
$\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \sqrt{5} \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
2016 AMC 12/AHSME, 20
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A?$
$\textbf{(A)}\ 385 \qquad
\textbf{(B)}\ 665 \qquad
\textbf{(C)}\ 945 \qquad
\textbf{(D)}\ 1140 \qquad
\textbf{(E)}\ 1330$
2017 AMC 10, 15
Rectangle $ABCD$ has $AB=3$ and $BC=4.$ Point $E$ is the foot of the perpendicular from $B$ to diagonal $\overline{AC}.$ What is the area of $\triangle ADE?$
$\textbf{(A)} \text{ 1} \qquad \textbf{(B)} \text{ }\frac{42}{25} \qquad \textbf{(C)} \text{ }\frac{28}{15} \qquad \textbf{(D)} \text{ 2} \qquad \textbf{(E)} \text{ }\frac{54}{25}$
2021 AMC 10 Fall, 19
Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. What is$$f(2) + f(3) + f(4) + f(5)+ f(6)?$$
$(\textbf{A})\: 8\qquad(\textbf{B}) \: 9\qquad(\textbf{C}) \: 11\qquad(\textbf{D}) \: 22\qquad(\textbf{E}) \: 29$
2018 AMC 10, 7
In the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the semicircles. The ratio $A:B$ is $1:18$. What is $N$?
[asy] draw((0,0)--(18,0)); draw(arc((9,0),9,0,180));
filldraw(arc((1,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((3,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((5,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((7,0),1,0,180)--cycle,gray(0.8)); label("...",(9,0.5)); filldraw(arc((11,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((13,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((15,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((17,0),1,0,180)--cycle,gray(0.8));
[/asy]
$\textbf{(A) } 16 \qquad \textbf{(B) } 17 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 36$
2017 AMC 10, 9
A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?
$\textbf{(A) } \frac{1}{27}\qquad \textbf{(B) } \frac{1}{9}\qquad \textbf{(C) } \frac{2}{9}\qquad \textbf{(D) } \frac{7}{27}\qquad \textbf{(E) } \frac{1}{2}$
2020 AMC 10, 15
Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions $2019, 2020, 2021$?
$\textbf{(A) } 7 \qquad\textbf{(B) } 9 \qquad\textbf{(C) } 10 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 12$
2016 AMC 10, 7
The ratio of the measures of two acute angles is $5:4$, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
$\textbf{(A)}\ 75\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 135\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 270$
2023 AMC 10, 20
Four congruent semicircles are drawn on the surface of a sphere with radius $2$, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curcve is $\pi \sqrt{n}$. What is $n$?
$\textbf{(A) } 32 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 48 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 27$
2021 AMC 10 Spring, 3
In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. among the 28 students in the program, 25% of the juniors and 10% of the seniors are on the debate team. how many juniors are in the program?
$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 20.$
2013 Stanford Mathematics Tournament, 3
Robin has obtained a circular pizza with radius $2$. However, being rebellious, instead of slicing the pizza radially, he decides to slice the pizza into $4$ strips of equal width both vertically and horizontally. What is the area of the smallest piece of pizza?
2020 AMC 10, 1
What is the value of $$1-(-2)-3-(-4)-5-(-6)?$$
$\textbf{(A) } -20 \qquad\textbf{(B) } -3 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 21$
2021 AMC 10 Spring, 19
Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer is $S$ is [i]also[/i] removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$?
$\textbf{(A)} ~36.2 \qquad\textbf{(B)} ~36.4 \qquad\textbf{(C)} ~36.6 \qquad\textbf{(D)} ~36.8 \qquad\textbf{(E)} ~37$
2016 AMC 10, 3
Let $x=-2016$. What is the value of $\left| \ \bigl \lvert { \ \lvert x\rvert -x }\bigr\rvert -|x|{\frac{}{}}^{}_{}\right|-x$?
$\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048$
2016 AMC 10, 4
Zoey read $15$ books, one at a time. The first book took her $1$ day to read, the second book took her $2$ days to read, the third book took her $3$ days to read, and so on, with each book taking her $1$ more day to read than the previous book. Zoey finished the first book on a monday, and the second on a Wednesday. On what day the week did she finish her $15$th book?
$\textbf{(A)}\ \text{Sunday}\qquad\textbf{(B)}\ \text{Monday}\qquad\textbf{(C)}\ \text{Wednesday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$
2021 AMC 12/AHSME Fall, 20
A cube is constructed from $4$ white unit cubes and $4$ black unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\
10 \qquad\textbf{(E)}\ 11$
2016 AMC 12/AHSME, 11
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$
$\textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 41 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 50 \qquad
\textbf{(E)}\ 57$
2021 AMC 12/AHSME Spring, 12
Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer is $S$ is [i]also[/i] removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$?
$\textbf{(A)} ~36.2 \qquad\textbf{(B)} ~36.4 \qquad\textbf{(C)} ~36.6 \qquad\textbf{(D)} ~36.8 \qquad\textbf{(E)} ~37$
2020 AMC 10, 6
Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome—it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this $2$-hour period?
$\textbf{(A) }50 \qquad \textbf{(B) }55 \qquad \textbf{(C) }60\qquad \textbf{(D) }65\qquad\textbf{(E) }70$
2021 AMC 10 Fall, 13
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]
$(\textbf{A})\: 19\frac14\qquad(\textbf{B}) \: 20\frac14\qquad(\textbf{C}) \: 21 \frac34\qquad(\textbf{D}) \: 22\frac12\qquad(\textbf{E}) \: 23\frac34$
2016 AMC 12/AHSME, 1
What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \frac{1}{2}$?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 20$
2021 AMC 10 Spring, 11
Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?
$(\textbf{A}) \: 24 \qquad (\textbf{B}) \: 30 \qquad (\textbf{C}) \: 48 \qquad (\textbf{D}) \: 60 \qquad (\textbf{E}) \: 64$
2019 AMC 10, 10
In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units?
$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$