Found problems: 3632
2023 AMC 12/AHSME, 5
Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3?
$\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{49}{216}\qquad\textbf{(C) }\frac{25}{108}\qquad\textbf{(D) }\frac{17}{72}\qquad\textbf{(E) }\frac{13}{54}$
2020 AMC 12/AHSME, 8
What is the median of the following list of $4040$ numbers$?$
$$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$$
$\textbf{(A) } 1974.5 \qquad \textbf{(B) } 1975.5 \qquad \textbf{(C) } 1976.5 \qquad \textbf{(D) } 1977.5 \qquad \textbf{(E) } 1978.5$
2008 AMC 12/AHSME, 7
For real numbers $ a$ and $ b$, define $ a\$b\equal{}(a\minus{}b)^2$. What is $ (x\minus{}y)^2\$(y\minus{}x)^2$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ x^2\plus{}y^2 \qquad
\textbf{(C)}\ 2x^2 \qquad
\textbf{(D)}\ 2y^2 \qquad
\textbf{(E)}\ 4xy$
1977 AMC 12/AHSME, 23
If the solutions of the equation $x^2+px+q=0$ are the cubes of the solutions of the equation $x^2+mx+n=0$, then
$\textbf{(A) }p=m^3+3mn\qquad\textbf{(B) }p=m^3-3mn\qquad$
$\textbf{(C) }p+q=m^3\qquad\textbf{(D) }\left(\frac{m}{n}\right)^2=\frac{p}{q}\qquad \textbf{(E) }\text{none of these}$
2017 AMC 12/AHSME, 3
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of these statements necessarily follows logically?
$\textbf{(A)}$ If Lewis did not receive an A, then he got all of the multiple choice questions wrong. \\
$\textbf{(B)}$ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong. \\
$\textbf{(C)}$ If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. \\
$\textbf{(D)}$ If Lewis received an A, then he got all of the multiple choice questions right. \\
$\textbf{(E)}$ If Lewis received an A, then he got at least one of the multiple choice questions right.
2018 AMC 10, 6
Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?
$\textbf{(A) } 200 \qquad \textbf{(B) } 300 \qquad \textbf{(C) } 400 \qquad \textbf{(D) } 500 \qquad \textbf{(E) } 600 $
2012 AMC 10, 4
Let $\angle ABC=24^\circ$ and $\angle ABD =20^\circ$. What is the smallest possible degree measure for $\angle CBD$?
$ \textbf{(A)}\ 0
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 12
$
2022 AMC 10, 5
Square $ABCD$ has side length $1$. Point $P$, $Q$, $R$, and $S$ each lie on a side of $ABCD$ such that $APQCRS$ is an equilateral convex hexagon with side length $s$. What is $s$?
$\textbf{(A) } \frac{\sqrt{2}}{3} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } 2-\sqrt{2} \qquad \textbf{(D) } 1-\frac{\sqrt{2}}{4} \qquad \textbf{(E) } \frac{2}{3}$
2009 AMC 10, 21
Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle, In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?
[asy]unitsize(6mm);
defaultpen(linewidth(.8pt));
draw(Circle((0,0),1+sqrt(2)));
draw(Circle((sqrt(2),0),1));
draw(Circle((0,sqrt(2)),1));
draw(Circle((-sqrt(2),0),1));
draw(Circle((0,-sqrt(2)),1));[/asy]$ \textbf{(A)}\ 3\minus{}2\sqrt2 \qquad
\textbf{(B)}\ 2\minus{}\sqrt2 \qquad
\textbf{(C)}\ 4(3\minus{}2\sqrt2) \qquad
\textbf{(D)}\ \frac12(3\minus{}\sqrt2)$
$ \textbf{(E)}\ 2\sqrt2\minus{}2$
2021 AMC 10 Spring, 23
A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square, The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}(a+b\sqrt{2}+\pi)$, where $a$ and $b$ are positive integers. What is $a+b$?
[asy]
//Diagram by Samrocksnature
draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0));
fill((2,0)--(0,2)--(0,0)--cycle, black);
fill((6,0)--(8,0)--(8,2)--cycle, black);
fill((8,6)--(8,8)--(6,8)--cycle, black);
fill((0,6)--(2,8)--(0,8)--cycle, black);
fill((4,6)--(2,4)--(4,2)--(6,4)--cycle, black);
filldraw(circle((2.6,3.31),0.47),gray);
[/asy]
$\textbf{(A) }64 \qquad \textbf{(B) }66 \qquad \textbf{(C) }68 \qquad \textbf{(D) }70 \qquad \textbf{(E) }72$
2009 AIME Problems, 15
Let $ \overline{MN}$ be a diameter of a circle with diameter $ 1$. Let $ A$ and $ B$ be points on one of the semicircular arcs determined by $ \overline{MN}$ such that $ A$ is the midpoint of the semicircle and $ MB\equal{}\frac35$. Point $ C$ lies on the other semicircular arc. Let $ d$ be the length of the line segment whose endpoints are the intersections of diameter $ \overline{MN}$ with the chords $ \overline{AC}$ and $ \overline{BC}$. The largest possible value of $ d$ can be written in the form $ r\minus{}s\sqrt{t}$, where $ r$, $ s$, and $ t$ are positive integers and $ t$ is not divisible by the square of any prime. Find $ r\plus{}s\plus{}t$.
1967 AMC 12/AHSME, 35
The roots of $64x^3-144x^2+92x-15=0$ are in arithmetic progression. The difference between the largest and smallest roots is:
$\textbf{(A)}\ 2\qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ \frac{1}{2}\qquad
\textbf{(D)}\ \frac{3}{8}\qquad
\textbf{(E)}\ \frac{1}{4}$
1986 India National Olympiad, 5
If $ P(x)$ is a polynomial with integer coefficients and $ a$, $ b$, $ c$, three distinct integers, then show that it is impossible to have $ P(a)\equal{}b$, $ P(b)\equal{}c$, $ P(c)\equal{}a$.
1959 AMC 12/AHSME, 12
By adding the same constant to $20,50,100$ a geometric progression results. The common ratio is:
$ \textbf{(A)}\ \frac53 \qquad\textbf{(B)}\ \frac43\qquad\textbf{(C)}\ \frac32\qquad\textbf{(D)}\ \frac12\qquad\textbf{(E)}\ \frac13 $
2010 AMC 10, 12
At the beginning of the school year, $ 50\%$ of all students in Mr. Well's math class answered "Yes" to the question "Do you love math", and $ 50\%$ answered "No." At the end of the school year, $ 70\%$ answered "Yes" and $ 30\%$ answered "No." Altogether, $ x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $ x$?
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 80$
2023 AMC 10, 16
Define an [i]upno[/i] to be a positive integer of $2$ or more digits where the digits are strictly increasing moving left to right. Similarly, define a [i]downno[/i] to be a positive integer of $2$ or more digits where the digits are strictly decreasing moving left to right. For instance, the number $258$ is an [i]upno[/i] and $8620$ is a [i]downno[/i]. Let $U$ equal the total number of [i]upno[/i]s and let $D$ equal the total number of [i]downno[/i]s. What is $|U-D|$?
$\textbf{(A)}~512\qquad\textbf{(B)}~10\qquad\textbf{(C)}~0\qquad\textbf{(D)}~9\qquad\textbf{(E)}~511$
2019 AMC 12/AHSME, 5
Two lines with slopes $\dfrac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10 ?$
$\textbf{(A) } 4 \qquad\textbf{(B) } 4\sqrt{2} \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 6\sqrt{2}$
2000 AIME Problems, 9
Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ,$ find the least integer that is greater than $z^{2000}+\frac 1{z^{2000}}.$
1994 AMC 12/AHSME, 29
Points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC, AB>r$, and the length of minor arc $BC$ is $r$. If angles are measured in radians, then $AB/BC=$
[asy]
draw(Circle((0,0), 13));
draw((-13,0)--(12,5)--(12,-5)--cycle);
dot((-13,0));
dot((12,5));
dot((12,-5));
label("A", (-13,0), W);
label("B", (12,5), NE);
label("C", (12,-5), SE);
[/asy]
$ \textbf{(A)}\ \frac{1}{2}\csc{\frac{1}{4}} \qquad\textbf{(B)}\ 2\cos{\frac{1}{2}} \qquad\textbf{(C)}\ 4\sin{\frac{1}{2}} \qquad\textbf{(D)}\ \csc{\frac{1}{2}} \qquad\textbf{(E)}\ 2\sec{\frac{1}{2}} $
2023 AMC 10, 18
Suppose $a$, $b$, and $c$ are positive integers such that \[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\] Which of the following statements are necessarily true?
I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$.
II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both.
III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$.
$\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$
2020 AMC 10, 2
Carl has $5$ cubes each having side length $1$, and Kate has $5$ cubes each having side length $2$. What is the total volume of the $10$ cubes?
$\textbf{(A) }24 \qquad \textbf{(B) }25 \qquad \textbf{(C) } 28\qquad \textbf{(D) } 40\qquad \textbf{(E) } 45$
1993 AMC 8, 6
A can of soup can feed $3$ adults or $5$ children. If there are $5$ cans of soup and $15$ children are fed, then how many adults would the remaining soup feed?
$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 10$
1967 AMC 12/AHSME, 40
Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$. To the nearest integer the area of triangle $ABC$ is:
$\textbf{(A)}\ 159\qquad
\textbf{(B)}\ 131\qquad
\textbf{(C)}\ 95\qquad
\textbf{(D)}\ 79\qquad
\textbf{(E)}\ 50$
2016 AMC 10, 25
Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \ge 0$?
$\textbf{(A)}\ 32\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ \text{infinitely many}$
2012 AMC 8, 5
In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is $X$, in centimeters?
[asy]
pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R;
A=(4,0);
B=(7,0);
C=(7,4);
D=(8,4);
E=(8,5);
F=(10,5);
G=(10,7);
H=(7,7);
I=(7,8);
J=(5,8);
K=(5,7);
L=(4,7);
M=(4,6);
N=(0,6);
O=(0,5);
P=(2,5);
Q=(2,3);
R=(4,3);
draw(A--B--C--D--E--F--G--H--I--J--K--L--M--N--O--P--Q--R--cycle);
label("$X$",(3.4,1.5));
label("6",(7.6,1.5));
label("1",(7.6,3.5));
label("1",(8.4,4.6));
label("2",(9.4,4.6));
label("2",(10.4,6));
label("3",(8.4,7.4));
label("1",(7.5,7.8));
label("2",(6,8.5));
label("1",(4.7,7.8));
label("1",(4.3,7.5));
label("1",(3.5,6.5));
label("4",(1.8,6.5));
label("1",(-0.5,5.5));
label("2",(0.8,4.5));
label("2",(1.5,3.8));
label("2",(2.8,2.6));
[/asy]
$\textbf{(A)}\hspace{.05in}1 \qquad \textbf{(B)}\hspace{.05in}2 \qquad \textbf{(C)}\hspace{.05in}3 \qquad \textbf{(D)}\hspace{.05in}4 \qquad \textbf{(E)}\hspace{.05in}5 $