Found problems: 3632
2015 AIME Problems, 4
Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$. Let $M$ be the midpoint of $\overline{AE}$, and $N$ be the midpoint of $\overline{CD}$. The area of $\triangle BMN$ is $x$. Find $x^2$.
1960 AMC 12/AHSME, 32
In this figure the center of the circle is $O$. $AB \perp BC$, $ADOE$ is a straight line, $AP = AD$, and $AB$ has a length twice the radius. Then:
[asy]
size(150);
defaultpen(linewidth(0.8)+fontsize(10));
real e=350,c=55;
pair O=origin,E=dir(e),C=dir(c),B=dir(180+c),D=dir(180+e), rot=rotate(90,B)*O,A=extension(E,D,B,rot);
path tangent=A--B;
pair P=waypoint(tangent,abs(A-D)/abs(A-B));
draw(unitcircle^^C--B--A--E);
dot(A^^B^^C^^D^^E^^P,linewidth(2));
label("$O$",O,dir(290));
label("$A$",A,N);
label("$B$",B,SW);
label("$C$",C,NE);
label("$D$",D,dir(120));
label("$E$",E,SE);
label("$P$",P,SW);[/asy]
$ \textbf{(A)} AP^2 = PB \times AB\qquad$
$\textbf{(B)}\ AP \times DO = PB \times AD\qquad$
$\textbf{(C)}\ AB^2 = AD \times DE\qquad$
$\textbf{(D)}\ AB \times AD = OB \times AO\qquad$
$\textbf{(E)}\ \text{none of these} $
2002 AMC 8, 10
$\textbf{Juan's Old Stamping Grounds}$
Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.)
[asy]
/* AMC8 2002 #8, 9, 10 Problem */
size(3inch, 1.5inch);
for ( int y = 0; y <= 5; ++y )
{
draw((0,y)--(18,y));
}
draw((0,0)--(0,5));
draw((6,0)--(6,5));
draw((9,0)--(9,5));
draw((12,0)--(12,5));
draw((15,0)--(15,5));
draw((18,0)--(18,5));
draw(scale(0.8)*"50s", (7.5,4.5));
draw(scale(0.8)*"4", (7.5,3.5));
draw(scale(0.8)*"8", (7.5,2.5));
draw(scale(0.8)*"6", (7.5,1.5));
draw(scale(0.8)*"3", (7.5,0.5));
draw(scale(0.8)*"60s", (10.5,4.5));
draw(scale(0.8)*"7", (10.5,3.5));
draw(scale(0.8)*"4", (10.5,2.5));
draw(scale(0.8)*"4", (10.5,1.5));
draw(scale(0.8)*"9", (10.5,0.5));
draw(scale(0.8)*"70s", (13.5,4.5));
draw(scale(0.8)*"12", (13.5,3.5));
draw(scale(0.8)*"12", (13.5,2.5));
draw(scale(0.8)*"6", (13.5,1.5));
draw(scale(0.8)*"13", (13.5,0.5));
draw(scale(0.8)*"80s", (16.5,4.5));
draw(scale(0.8)*"8", (16.5,3.5));
draw(scale(0.8)*"15", (16.5,2.5));
draw(scale(0.8)*"10", (16.5,1.5));
draw(scale(0.8)*"9", (16.5,0.5));
label(scale(0.8)*"Country", (3,4.5));
label(scale(0.8)*"Brazil", (3,3.5));
label(scale(0.8)*"France", (3,2.5));
label(scale(0.8)*"Peru", (3,1.5));
label(scale(0.8)*"Spain", (3,0.5));
label(scale(0.9)*"Juan's Stamp Collection", (9,0), S);
label(scale(0.9)*"Number of Stamps by Decade", (9,5), N);
[/asy]
The average price of his '70s stamps is closest to
$\text{(A)}\ 3.5 \text{ cents} \qquad \text{(B)}\ 4 \text{ cents} \qquad \text{(C)}\ 4.5 \text{ cents} \qquad \text{(D)}\ 5 \text{ cents} \qquad \text{(E)}\ 5.5 \text{ cents}$
2014 USAMTS Problems, 3a:
A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in almost-order with heights (from left-to-right) of $160, 170, 165, 180, 175$ centimeters.
(a) How many different ways are there to line up $10$ people in [i]almost-order[/i] if their heights are $140, 145, 150, 155,$ $160,$ $165,$ $170,$ $175,$ $180$, and $185$ centimeters?
2012 AMC 12/AHSME, 9
A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$) or is divisible by $4$ but not by $100$ (such as $2012$). The $200\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born?
$ \textbf{(A)}\ \text{Friday}
\qquad\textbf{(B)}\ \text{Saturday}
\qquad\textbf{(C)}\ \text{Sunday}
\qquad\textbf{(D)}\ \text{Monday}
\qquad\textbf{(E)}\ \text{Tuesday}
$
2020 AMC 10, 18
An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
$\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12$
2013 AMC 12/AHSME, 17
A group of $ 12 $ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $ k^\text{th} $ pirate to take a share takes $ \frac{k}{12} $ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $ 12^{\text{th}} $ pirate receive?
$ \textbf{(A)} \ 720 \qquad \textbf{(B)} \ 1296 \qquad \textbf{(C)} \ 1728 \qquad \textbf{(D)} \ 1925 \qquad \textbf{(E)} \ 3850 $
2015 AMC 12/AHSME, 13
A league with $12$ teams holds a round-robin tournament, with each team playing every other team once. Games either end with one team victorious or else end in a draw. A team scores $2$ points for every game it wins and $1$ point for every game it draws. Which of the following is $\textbf{not}$ a true statement about the list of $12$ scores?
$\textbf{(A) }\text{There must be an even number of odd scores.}$
$\textbf{(B) }\text{There must be an even number of even scores.}$
$\textbf{(C) }\text{There cannot be two scores of 0.}$
$\textbf{(D) }\text{The sum of the scores must be at least 100.}$
$\textbf{(E) }\text{The highest score must be at least 12.}$
2000 AMC 10, 10
The sides of a triangle with positive area have lengths $ 4$, $ 6$, and $ x$. The sides of a second triangle with positive area have lengths $ 4$, $ 6$, and $ y$. What is the smallest positive number that is [b]not[/b] a possible value of $ |x \minus{} y|$?
$ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 10$
2009 AMC 12/AHSME, 16
Trapezoid $ ABCD$ has $ AD\parallel{}BC$, $ BD \equal{} 1$, $ \angle DBA \equal{} 23^{\circ}$, and $ \angle BDC \equal{} 46^{\circ}$. The ratio $ BC: AD$ is $ 9: 5$. What is $ CD$?
$ \textbf{(A)}\ \frac {7}{9}\qquad \textbf{(B)}\ \frac {4}{5}\qquad \textbf{(C)}\ \frac {13}{15} \qquad \textbf{(D)}\ \frac {8}{9}\qquad \textbf{(E)}\ \frac {14}{15}$
2009 AMC 12/AHSME, 10
In quadrilateral $ ABCD$, $ AB \equal{} 5$, $ BC \equal{} 17$, $ CD \equal{} 5$, $ DA \equal{} 9$, and $ BD$ is an integer. What is $ BD$?
[asy]unitsize(4mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair C=(0,0), B=(17,0);
pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0];
pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0];
pair[] dotted={A,B,C,D};
draw(D--A--B--C--D--B);
dot(dotted);
label("$D$",D,NW);
label("$C$",C,W);
label("$B$",B,E);
label("$A$",A,NE);[/asy]$ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$
2012 AMC 10, 10
Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
$ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 $
2013 AMC 10, 3
On a particular January day, the high temperature in Lincoln, Nebraska, was 16 degrees higher than the low temperature, and the average of the high and low temperatures was $3^{\circ}$. In degrees, what was the low temperature in Lincoln that day?
$\textbf{(A) }-13\qquad\textbf{(B) }-8\qquad\textbf{(C) }-5\qquad\textbf{(D) }3\qquad\textbf{(E) }11$
2011 AMC 10, 16
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?
[asy]
unitsize(10mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;
pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2));
draw(A--B--C--D--E--F--G--H--cycle);
draw(A--D);
draw(B--G);
draw(C--F);
draw(E--H);
[/asy]
$ \textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad\textbf{(E)}\ 2 - \sqrt{2}$
1983 AMC 12/AHSME, 27
A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.)
$ \textbf{(A)}\ \frac{5}{2}\qquad\textbf{(B)}\ 9 - 4\sqrt{5}\qquad\textbf{(C)}\ 8\sqrt{10} - 23\qquad\textbf{(D)}\ 6 - \sqrt{15}\qquad\textbf{(E)}\ 10\sqrt{5} - 20 $
1963 AMC 12/AHSME, 40
If $x$ is a number satisfying the equation $\sqrt[3]{x+9}-\sqrt[3]{x-9}=3$, then $x^2$ is between:
$\textbf{(A)}\ 55\text{ and }65 \qquad
\textbf{(B)}\ 65\text{ and }75\qquad
\textbf{(C)}\ 75\text{ and }85 \qquad
\textbf{(D)}\ 85\text{ and }95 \qquad
\textbf{(E)}\ 95\text{ and }105$
2012 AMC 12/AHSME, 19
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?
$ \textbf{(A)}\ 60
\qquad\textbf{(B)}\ 170
\qquad\textbf{(C)}\ 290
\qquad\textbf{(D)}\ 320
\qquad\textbf{(E)}\ 660
$
1990 AMC 12/AHSME, 7
A triangle with integral sides has perimeter $8$. The area of the triangle is
$\textbf{(A) }2\sqrt{2}\qquad
\textbf{(B) }\dfrac{16}{9}\sqrt{3}\qquad
\textbf{(C) }2\sqrt{3}\qquad
\textbf{(D) }4\qquad
\textbf{(E) }4\sqrt{2}$
2014 AMC 10, 24
A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence?
$ \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996507\qquad\textbf{(C)}\ 996508\qquad\textbf{(D)}\ 996509\qquad\textbf{(E)}\ 996510 $
1967 AMC 12/AHSME, 37
Segments $AD=10$, $BE=6$, $CF=24$ are drawn from the vertices of triangle $ABC$, each perpendicular to a straight line $RS$, not intersecting the triangle. Points $D$, $E$, $F$ are the intersection points of $RS$ with the perpendiculars. If $x$ is the length of the perpendicular segment $GH$ drawn to $RS$ from the intersection point $G$ of the medians of the triangle, then $x$ is:
$\textbf{(A)}\ \frac{40}{3}\qquad
\textbf{(B)}\ 16\qquad
\textbf{(C)}\ \frac{56}{3}\qquad
\textbf{(D)}\ \frac{80}{3}\qquad
\textbf{(E)}\ \text{undetermined}$
2014 AIME Problems, 11
A token starts at the point $(0,0)$ of an $xy$-coordinate grid and them makes a sequence of six moves. Each move is $1$ unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of $|y|=|x|$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2010 AIME Problems, 13
The $ 52$ cards in a deck are numbered $ 1, 2, \ldots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let $ p(a)$ be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards $ a$ and $ a\plus{}9$, and Dylan picks the other of these two cards. The minimum value of $ p(a)$ for which $ p(a)\ge\frac12$ can be written as $ \frac{m}{n}$. where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
1995 AIME Problems, 4
Circles of radius 3 and 6 are externally tangent to each other and are internally tangent to a circle of radius 9. The circle of radius 9 has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.
2012 AMC 12/AHSME, 20
A trapezoid has side lengths $3, 5, 7,$ and $11$. The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \sqrt{n_1} + r_2 \sqrt{n_2} + r_3$, where $r_1, r_2,$ and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of a prime. What is the greatest integer less than or equal to
\[r_1 + r_2 + r_3 + n_1 + n_2?\]
$ \textbf{(A)}\ 57\qquad\textbf{(B)}\ 59\qquad\textbf{(C)}\ 61\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65 $
2014 AMC 8, 25
A straight one-mile stretch of highway, $40$ feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at $5$ miles per hour, how many hours will it take to cover the one-mile stretch?
Note: $1$ mile= $5280$ feet
[asy]size(10cm); pathpen=black; pointpen=black;
D(arc((-2,0),1,300,360));
D(arc((0,0),1,0,180));
D(arc((2,0),1,180,360));
D(arc((4,0),1,0,180));
D(arc((6,0),1,180,240));
D((-1.5,1)--(5.5,1));
D((-1.5,0)--(5.5,0),dashed);
D((-1.5,-1)--(5.5,-1));
[/asy]
$\textbf{(A) }\frac{\pi}{11}\qquad\textbf{(B) }\frac{\pi}{10}\qquad\textbf{(C) }\frac{\pi}{5}\qquad\textbf{(D) }\frac{2\pi}{5}\qquad \textbf{(E) }\frac{2\pi}{3}$