Found problems: 730
2024 AMC 10, 21
Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?
[asy]
size(6cm);
draw(circle((0,1),1), linewidth(1.2));
draw((-1,0)--(1.25,0), linewidth(1.2));
draw(circle((1,1/4),1/4), linewidth(1.2));
[/asy]
$\textbf{(A)}~\displaystyle\frac{1}{9}
\qquad\textbf{(B)}~1
\qquad\textbf{(C)}~\displaystyle\frac{10}{9}
\qquad\textbf{(D)}~\displaystyle\frac{11}{9}
\qquad\textbf{(E)}~\displaystyle\frac{19}{9}$
2024 AMC 10, 8
Let $N$ be the product of all the positive integer divisors of $42$. What is the units digit of $N$?
$
\textbf{(A) }0 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }4 \qquad
\textbf{(D) }6 \qquad
\textbf{(E) }8 \qquad
$
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$$f(2) + f(3) + f(4) + f(5)+ f(6)?$$
$(\textbf{A})\: 8\qquad(\textbf{B}) \: 9\qquad(\textbf{C}) \: 11\qquad(\textbf{D}) \: 22\qquad(\textbf{E}) \: 29$
2016 AMC 10, 5
A rectangular box has integer side lengths in the ratio $1: 3: 4$. Which of the following could be the volume of the box?
$\textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 144$
2022 AMC 10, 22
Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$?
$\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad$
2023 AMC 10, 10
Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1$. If she scores an $11$ on each of the next three quizzes, her mean will increase by $2$. What is the mean of her quiz scores currently?
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$
2014 AMC 10, 5
On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}}\ 4\qquad\textbf{(E)}\ 5$
2023 AMC 10, 24
What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w,v+4w)$ with $0 \le u \le 1$, $0 \le v \le 1$, and $0 \le w \le 1$? \\ \\
$\textbf{(A) } 10\sqrt{3} \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 16$
2022 AMC 10, 1
Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y$. What is the value of \[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\]
$ \textbf{(A)}\ -2 \qquad
\textbf{(B)}\ -1 \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ 2$
2019 AMC 10, 16
The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1 ?$
[asy]unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);[/asy]
$\textbf{(A) } 4 \pi \sqrt{3} \qquad\textbf{(B) } 7 \pi \qquad\textbf{(C) } \pi(3\sqrt{3} +2) \qquad\textbf{(D) } 10 \pi (\sqrt{3} - 1) \qquad\textbf{(E) } \pi(\sqrt{3} + 6)$
2017 AMC 10, 22
The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and the line $ED$ is perpendicular to the line $AD$. Segment $\overline{AE}$ intersects the circle at point $C$ between $A$ and $E$. What is the area of $\triangle ABC$?
$\textbf{(A) \ } \frac{120}{37}\qquad \textbf{(B) \ } \frac{140}{39}\qquad \textbf{(C) \ } \frac{145}{39}\qquad \textbf{(D) \ } \frac{140}{37}\qquad \textbf{(E) \ } \frac{120}{31}$
2021 AMC 10 Spring, 22
Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly $19$. How many sheets were borrowed?
$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 20$
2022 AMC 12/AHSME, 4
The least common multiple of a positive integer $n$ and 18 is 180, and the greatest common divisor of $n$ and 45 is 15. What is the sum of the digits of $n$?
$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }12$
2024 AMC 10, 18
How many different remainders can result when the $100$th power of an integer is divided by $125$?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }5 \qquad
\textbf{(D) }25 \qquad
\textbf{(E) }125 \qquad
$
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]
$\textbf{(A) } 16 \qquad \textbf{(B) } 17 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 36$
2022 AMC 10, 4
A donkey suffers an attack of hiccups and the first hiccup happens at $\text{4:00}$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. At what time does the donkey’s $\text{700th}$ hiccup occur?
$\textbf{(A) }$ $15$ seconds after $\text{4:58}$
$\textbf{(B) }$ $20$ seconds after $\text{4:58}$
$\textbf{(C)}$ $25$ seconds after $\text{4:58}$
$\textbf{(D) }$ $30$ seconds after $\text{4:58}$
$\textbf{(E) }$ $35$ seconds after $\text{4:58}$
2010 AMC 10, 19
Equiangular hexagon $ ABCDEF$ has side lengths $ AB \equal{} CD \equal{} EF \equal{} 1$ and $ BC \equal{} DE \equal{} FA \equal{} r$. The area of $ \triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $ r$?
$ \textbf{(A)}\ \frac {4\sqrt {3}}{3} \qquad
\textbf{(B)}\ \frac {10}{3} \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ \frac {17}{4} \qquad
\textbf{(E)}\ 6$
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?
$\textbf{(A) } \frac{1}{27}\qquad \textbf{(B) } \frac{1}{9}\qquad \textbf{(C) } \frac{2}{9}\qquad \textbf{(D) } \frac{7}{27}\qquad \textbf{(E) } \frac{1}{2}$
2013 AMC 10, 15
Two sides of a triangle have lengths $10$ and $15$. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side?
$\textbf{(A) }6\qquad
\textbf{(B) }8\qquad
\textbf{(C) }9\qquad
\textbf{(D) }12\qquad
\textbf{(E) }18\qquad$
2013 Purple Comet Problems, 28
Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \times30\times30$ cube as shown. The two figures $ACFH$ and $BDEG$ are congruent regular tetrahedra. Find the volume of the intersection of these two tetrahedra.
[asy]
import graph; size(12.57cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
pen dotstyle = black;
real xmin = -3.79, xmax = 8.79, ymin = 0.32, ymax = 4.18; /* image dimensions */
pen ffqqtt = rgb(1,0,0.2); pen ffzzzz = rgb(1,0.6,0.6); pen zzzzff = rgb(0.6,0.6,1);
draw((6,3.5)--(8,1.5), zzzzff);
draw((7,3)--(5,1), blue);
draw((6,3.5)--(7,3), blue);
draw((6,3.5)--(5,1), blue);
draw((5,1)--(8,1.5), blue);
draw((7,3)--(8,1.5), blue);
draw((4,3.5)--(2,1.5), ffzzzz);
draw((1,3)--(2,1.5), ffqqtt);
draw((2,1.5)--(3,1), ffqqtt);
draw((1,3)--(3,1), ffqqtt);
draw((4,3.5)--(1,3), ffqqtt);
draw((4,3.5)--(3,1), ffqqtt);
draw((-3,3)--(-3,1), linewidth(1.6));
draw((-3,3)--(-1,3), linewidth(1.6));
draw((-1,3)--(-1,1), linewidth(1.6));
draw((-3,1)--(-1,1), linewidth(1.6));
draw((-3,3)--(-2,3.5), linewidth(1.6));
draw((-2,3.5)--(0,3.5), linewidth(1.6));
draw((0,3.5)--(-1,3), linewidth(1.6));
draw((0,3.5)--(0,1.5), linewidth(1.6));
draw((0,1.5)--(-1,1), linewidth(1.6));
draw((-3,1)--(-2,1.5));
draw((-2,1.5)--(0,1.5));
draw((-2,3.5)--(-2,1.5));
draw((1,3)--(1,1), linewidth(1.6));
draw((1,3)--(3,3), linewidth(1.6));
draw((3,3)--(3,1), linewidth(1.6));
draw((1,1)--(3,1), linewidth(1.6));
draw((1,3)--(2,3.5), linewidth(1.6));
draw((2,3.5)--(4,3.5), linewidth(1.6));
draw((4,3.5)--(3,3), linewidth(1.6));
draw((4,3.5)--(4,1.5), linewidth(1.6));
draw((4,1.5)--(3,1), linewidth(1.6));
draw((1,1)--(2,1.5));
draw((2,3.5)--(2,1.5));
draw((2,1.5)--(4,1.5));
draw((5,3)--(5,1), linewidth(1.6));
draw((5,3)--(6,3.5), linewidth(1.6));
draw((5,3)--(7,3), linewidth(1.6));
draw((7,3)--(7,1), linewidth(1.6));
draw((5,1)--(7,1), linewidth(1.6));
draw((6,3.5)--(8,3.5), linewidth(1.6));
draw((7,3)--(8,3.5), linewidth(1.6));
draw((7,1)--(8,1.5));
draw((5,1)--(6,1.5));
draw((6,3.5)--(6,1.5));
draw((6,1.5)--(8,1.5));
draw((8,3.5)--(8,1.5), linewidth(1.6));
label("$ A $",(-3.4,3.41),SE*labelscalefactor);
label("$ D $",(-2.16,4.05),SE*labelscalefactor);
label("$ H $",(-2.39,1.9),SE*labelscalefactor);
label("$ E $",(-3.4,1.13),SE*labelscalefactor);
label("$ F $",(-1.08,0.93),SE*labelscalefactor);
label("$ G $",(0.12,1.76),SE*labelscalefactor);
label("$ B $",(-0.88,3.05),SE*labelscalefactor);
label("$ C $",(0.17,3.85),SE*labelscalefactor);
label("$ A $",(0.73,3.5),SE*labelscalefactor);
label("$ B $",(3.07,3.08),SE*labelscalefactor);
label("$ C $",(4.12,3.93),SE*labelscalefactor);
label("$ D $",(1.69,4.07),SE*labelscalefactor);
label("$ E $",(0.60,1.15),SE*labelscalefactor);
label("$ F $",(2.96,0.95),SE*labelscalefactor);
label("$ G $",(4.12,1.67),SE*labelscalefactor);
label("$ H $",(1.55,1.82),SE*labelscalefactor);
label("$ A $",(4.71,3.47),SE*labelscalefactor);
label("$ B $",(7.14,3.10),SE*labelscalefactor);
label("$ C $",(8.14,3.82),SE*labelscalefactor);
label("$ D $",(5.78,4.08),SE*labelscalefactor);
label("$ E $",(4.6,1.13),SE*labelscalefactor);
label("$ F $",(6.93,0.96),SE*labelscalefactor);
label("$ G $",(8.07,1.64),SE*labelscalefactor);
label("$ H $",(5.65,1.90),SE*labelscalefactor);
dot((-3,3),dotstyle);
dot((-3,1),dotstyle);
dot((-1,3),dotstyle);
dot((-1,1),dotstyle);
dot((-2,3.5),dotstyle);
dot((0,3.5),dotstyle);
dot((0,1.5),dotstyle);
dot((-2,1.5),dotstyle);
dot((1,3),dotstyle);
dot((1,1),dotstyle);
dot((3,3),dotstyle);
dot((3,1),dotstyle);
dot((2,3.5),dotstyle);
dot((4,3.5),dotstyle);
dot((4,1.5),dotstyle);
dot((2,1.5),dotstyle);
dot((5,3),dotstyle);
dot((5,1),dotstyle);
dot((6,3.5),dotstyle);
dot((7,3),dotstyle);
dot((7,1),dotstyle);
dot((8,3.5),dotstyle);
dot((8,1.5),dotstyle);
dot((6,1.5),dotstyle); [/asy]
2018 AMC 10, 19
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
$\textbf{(A) } 7 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11 $
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 $2019, 2020, 2021$?
$\textbf{(A) } 7 \qquad\textbf{(B) } 9 \qquad\textbf{(C) } 10 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 12$
2017 AMC 12/AHSME, 18
The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and the line $ED$ is perpendicular to the line $AD$. Segment $\overline{AE}$ intersects the circle at point $C$ between $A$ and $E$. What is the area of $\triangle ABC$?
$\textbf{(A) \ } \frac{120}{37}\qquad \textbf{(B) \ } \frac{140}{39}\qquad \textbf{(C) \ } \frac{145}{39}\qquad \textbf{(D) \ } \frac{140}{37}\qquad \textbf{(E) \ } \frac{120}{31}$
2020 AMC 10, 15
A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 23$
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?
$\textbf{(A)}\ 75\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 135\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 270$