Found problems: 649
2021 AMC 12/AHSME Spring, 20
Suppose that on a parabola with vertex $V$ and a focus $F$ there exists a point $A$ such that $AF=20$ and $AV=21$. What is the sum of all possible values of the length $FV?$
$\textbf{(A) }13 \qquad \textbf{(B) }\frac{40}3 \qquad \textbf{(C) }\frac{41}3 \qquad \textbf{(D) }14\qquad \textbf{(E) }\frac{43}3$
Proposed by [b]djmathman[/b]
2019 AMC 10, 23
Points $A(6,13)$ and $B(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$?
$\textbf{(A) }\frac{83\pi}{8}\qquad\textbf{(B) }\frac{21\pi}{2}\qquad\textbf{(C) }
\frac{85\pi}{8}\qquad\textbf{(D) }\frac{43\pi}{4}\qquad\textbf{(E) }\frac{87\pi}{8}$
2020 AMC 10, 23
Let $T$ be the triangle in the coordinate plane with vertices $\left(0,0\right)$, $\left(4,0\right)$, and $\left(0,3\right)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)
$\textbf{(A) } 12\qquad\textbf{(B) } 15\qquad\textbf{(C) }17 \qquad\textbf{(D) }20 \qquad\textbf{(E) }25$
2024 AMC 10, 4
Balls numbered $1,2,3,\ldots$ are deposited in $5$ bins, labeled $A,B,C,D,$ and $E$, using the following procedure. Ball $1$ is deposited in bin $A$, and balls $2$ and $3$ are deposted in $B$. The next three balls are deposited in bin $C$, the next $4$ in bin $D$, and so on, cycling back to bin $A$ after balls are deposited in bin $E$. (For example, $22,23,\ldots,28$ are despoited in bin $B$ at step 7 of this process.) In which bin is ball $2024$ deposited?
$\textbf{(A) }A\qquad\textbf{(B) }B\qquad\textbf{(C) }C\qquad\textbf{(D) }D\qquad\textbf{(E) }E$
2021 AMC 12/AHSME Fall, 17
How many ordered pairs of positive integers $(b,c)$ exist where both $x^2+bx+c=0$ and $x^2+cx+b=0$ do not have distinct, real solutions?
$\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 12 \qquad$
2016 AMC 12/AHSME, 16
The graphs of $y=\log_3x$, $y=\log_x3$, $y=\log_{\frac13}x$, and $y=\log_x\frac13$ are plotted on the same set of axes. How many points in the plane with positive $x-$coordinates lie on two or more of the graphs?
$\textbf{(A) } 2 \qquad\textbf{(B) } 3 \qquad\textbf{(C) } 4 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 6$
2022 AMC 12/AHSME, 14
The graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. What is $\tan(\angle ABC)$?
$\textbf{(A)}\frac{1}{7}~\textbf{(B)}\frac{1}{4}~\textbf{(C)}\frac{3}{7}~\textbf{(D)}\frac{1}{2}~\textbf{(E)}\frac{4}{7}$
2021 AMC 12/AHSME Fall, 15
Recall that the conjugate of the complex number $w = a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$, is the complex number $\overline{w} = a - bi$. For any complex number $z$, let $f(z) = 4i\hspace{1pt}\overline{z}$. The polynomial $P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$ has four complex roots: $z_1$, $z_2$, $z_3$, and $z_4$. Let $Q(z) = z^4 + Az^3 + Bz^2 + Cz + D$ be the polynomial whose roots are $f(z_1)$, $f(z_2)$, $f(z_3)$, and $f(z_4)$, where the coefficients $A,$ $B,$ $C,$ and $D$ are complex numbers. What is $B + D?$
$(\textbf{A})\: {-}304\qquad(\textbf{B}) \: {-}208\qquad(\textbf{C}) \: 12i\qquad(\textbf{D}) \: 208\qquad(\textbf{E}) \: 304$
2021 AMC 12/AHSME Spring, 9
Which of the following is equivalent to $$(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$$
$\textbf{(A) }3^{127}+2^{127} \qquad \textbf{(B) }3^{127}+2^{127}+2\cdot 3^{63}+3\cdot 2^{63} \qquad \textbf{(C) }3^{128}-2^{128} \qquad \textbf{(D) }3^{128}+2^{128} \qquad \textbf{(E) }5^{127}$
2023 AMC 10, 7
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?
$\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{49}{216}\qquad\textbf{(C) }\frac{25}{108}\qquad\textbf{(D) }\frac{17}{72}\qquad\textbf{(E) }\frac{13}{54}$
2023 AMC 12/AHSME, 15
Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta=\angle PAB=\angle QPC=\angle RQB=\cdots$ will produce in a length that is $120$ meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.)
[asy]
import olympiad;
draw((-50,15)--(50,15));
draw((50,15)--(50,-15));
draw((50,-15)--(-50,-15));
draw((-50,-15)--(-50,15));
draw((-50,-15)--(-22.5,15));
draw((-22.5,15)--(5,-15));
draw((5,-15)--(32.5,15));
draw((32.5,15)--(50,-4.090909090909));
label("$\theta$", (-41.5,-10.5));
label("$\theta$", (-13,10.5));
label("$\theta$", (15.5,-10.5));
label("$\theta$", (43,10.5));
dot((-50,15));
dot((-50,-15));
dot((50,15));
dot((50,-15));
dot((50,-4.09090909090909));
label("$D$",(-58,15));
label("$A$",(-58,-15));
label("$C$",(58,15));
label("$B$",(58,-15));
label("$S$",(58,-4.0909090909));
dot((-22.5,15));
dot((5,-15));
dot((32.5,15));
label("$P$",(-22.5,23));
label("$Q$",(5,-23));
label("$R$",(32.5,23));
[/asy]
$\textbf{(A)}~\arccos\frac{5}{6}\qquad\textbf{(B)}~\arccos\frac{4}{5}\qquad\textbf{(C)}~\arccos\frac{3}{10}\qquad\textbf{(D)}~\arcsin\frac{4}{5}\qquad\textbf{(E)}~\arcsin\frac{5}{6}$
2024 AMC 10, 2
What is $10! - 7! \cdot 6!$?
$
\textbf{(A) }-120 \qquad
\textbf{(B) }0 \qquad
\textbf{(C) }120 \qquad
\textbf{(D) }600 \qquad
\textbf{(E) }720 \qquad
$
2018 AMC 12/AHSME, 9
What is \[ \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ? \]
$
\textbf{(A) }100,100 \qquad
\textbf{(B) }500,500\qquad
\textbf{(C) }505,000 \qquad
\textbf{(D) }1,001,000 \qquad
\textbf{(E) }1,010,000 \qquad
$
1959 AMC 12/AHSME, 30
$A$ can run around a circular track in $40$ seconds. $B$, running in the opposite direction, meets $A$ every $15$ seconds. What is $B$'s time to run around the track, expressed in seconds?
$ \textbf{(A)}\ 12\frac12 \qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 27\frac12\qquad\textbf{(E)}\ 55 $
2023 AMC 12/AHSME, 24
Integers $a, b, c, d$ satisfy the following:
$abcd=2^6\cdot 3^9\cdot 5^7$
$\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3$
$\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3$
$\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3$
$\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2$
$\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2$
$\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2$
Find $\text{gcd}(a,b,c,d)$
$\textbf{(A)}~30\qquad\textbf{(B)}~45\qquad\textbf{(C)}~3\qquad\textbf{(D)}~15\qquad\textbf{(E)}~6$
2017 AMC 12/AHSME, 6
Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$
1959 AMC 12/AHSME, 17
If $y=a+\frac{b}{x}$, where $a$ and $b$ are constants, and if $y=1$ when $x=-1$, and $y=5$ when $x=-5$, then $a+b$ equals:
$ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 $
2017 AMC 10, 11
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
$\textbf{(A) } 10\%\qquad \textbf{(B) } 12\%\qquad \textbf{(C) } 20\%\qquad \textbf{(D) } 25\%\qquad \textbf{(E) } 33\frac{1}{3}\%$
2012 AMC 12/AHSME, 21
Let $a,b,$ and $c$ be positive integers with $a\ge b\ge c$ such that
\begin{align*} a^2-b^2-c^2+ab&=2011\text{ and}\\
a^2+3b^2+3c^2-3ab-2ac-2bc&=-1997\end{align*}
What is $a$?
$ \textbf{(A)}\ 249
\qquad\textbf{(B)}\ 250
\qquad\textbf{(C)}\ 251
\qquad\textbf{(D)}\ 252
\qquad\textbf{(E)}\ 253
$
2020 AMC 10, 16
A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0),$ $(2020, 0),$ $(2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth$?$
$\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7$
2012 AMC 12/AHSME, 16
Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?
$ \textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqrt{7}\qquad\textbf{(E)}\ \sqrt{30} $
2019 AMC 12/AHSME, 15
As shown in the figure, line segment $\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=2.$ Three semicircles of radius $1,$ $\overarc{AEB},\overarc{BFC},$ and $\overarc{CGD},$ have their diameters on $\overline{AD},$ and are tangent to line $EG$ at $E,F,$ and $G,$ respectively. A circle of radius $2$ has its center on $F. $ The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form
\[\frac{a}{b}\cdot\pi-\sqrt{c}+d,\]
where $a,b,c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d$?
[asy]
size(6cm);
filldraw(circle((0,0),2), gray(0.7));
filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0));
filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0));
filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0));
dot((-3,-1));
label("$A$",(-3,-1),S);
dot((-2,0));
label("$E$",(-2,0),NW);
dot((-1,-1));
label("$B$",(-1,-1),S);
dot((0,0));
label("$F$",(0,0),N);
dot((1,-1));
label("$C$",(1,-1), S);
dot((2,0));
label("$G$", (2,0),NE);
dot((3,-1));
label("$D$", (3,-1), S);
[/asy]
$\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16\qquad\textbf{(E) } 17$
2018 AMC 10, 2
Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes?
$\textbf{(A) } 64 \qquad \textbf{(B) } 65 \qquad \textbf{(C) } 66 \qquad \textbf{(D) } 67 \qquad \textbf{(E) } 68$
2023 AMC 12/AHSME, 16
In Coinland, there are three types of coins, each worth $6,$ $10,$ and $15.$ What is the sum of the digits of the maximum amount of money that is impossible to have?
$\textbf{(A) }11\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$
(I forgot the order)
2021 AMC 12/AHSME Fall, 12
For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n$. For example,
\[f(14) = (1 + 2 + 7 + 14) \div 14 = \frac{12}{7}.\]
What is $f(768) - f(384)?$
$\textbf{(A) }\frac{1}{768}\qquad\textbf{(B) }\frac{1}{192}\qquad\textbf{(C) }1\qquad\textbf{(D) }\frac{4}{3}\qquad\textbf{(E) }\frac{8}{3}$