Found problems: 3632
2008 AMC 10, 24
Quadrilateral $ABCD$ has $AB=BC=CD$, $\angle ABC=70^\circ$, and $\angle BCD=170^\circ$. What is the degree measure of $\angle BAD$?
$ \textbf{(A)}\ 75\qquad
\textbf{(B)}\ 80\qquad
\textbf{(C)}\ 85\qquad
\textbf{(D)}\ 90\qquad
\textbf{(E)}\ 95$
1972 AMC 12/AHSME, 13
[asy]
draw(unitsquare);draw((0,0)--(.4,1)^^(0,.6)--(1,.2));
label("D",(0,1),NW);label("E",(.4,1),N);label("C",(1,1),NE);
label("P",(0,.6),W);label("M",(.25,.55),E);label("Q",(1,.2),E);
label("A",(0,0),SW);label("B",(1,0),SE);
//Credit to Zimbalono for the diagram[/asy]
Inside square $ABCD$ (See figure) with sides of length $12$ inches, segment $AE$ is drawn where $E$ is the point on $DC$ which is $5$ inches from $D$. The perpendicular bisector of $AE$ is drawn and intersects $AE$, $AD$, and $BC$ at points $M$, $P$, and $Q$ respectively. The ratio of segment $PM$ to $MQ$ is
$\textbf{(A) }5:12\qquad\textbf{(B) }5:13\qquad\textbf{(C) }5:19\qquad\textbf{(D) }1:4\qquad \textbf{(E) }5:21$
2011 AIME Problems, 10
The probability that a set of three distinct vertices chosen at random from among the vertices of a regular $n$-gon determine an obtuse triangle is $\tfrac{93}{125}$. Find the sum of all possible values of $n$.
2018 AIME Problems, 15
David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, \(A\), \(B\), \(C\), which can each be inscribed in a circle with radius \(1\). Let \(\varphi_A\) denote the measure of the acute angle made by the diagonals of quadrilateral \(A\), and define \(\varphi_B\) and \(\varphi_C\) similarly. Suppose that \(\sin\varphi_A=\frac{2}{3}\), \(\sin\varphi_B=\frac{3}{5}\), and \(\sin\varphi_C=\frac{6}{7}\). All three quadrilaterals have the same area \(K\), which can be written in the form \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).
2009 AIME Problems, 9
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $ \$1$ to $ \$9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were $ 1, 1, 1, 1, 3, 3, 3$. Find the total number of possible guesses for all three prizes consistent with the hint.
2021 AMC 10 Spring, 4
A cart rolls down a hill, traveling 5 inches the first second and accelerating so that each successive 1-second time
interval, it travels 7 inches more than during the previous 1-second interval. The cart takes 30 seconds to reach the
bottom of the hill. How far, in inches, does it travel?
$\textbf{(A) }215 \qquad \textbf{(B) }360 \qquad \textbf{(C) }2992 \qquad \textbf{(D) }3195 \qquad \textbf{(E) }3242$
2014 AMC 12/AHSME, 17
A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$?
[asy]
import graph3;
import solids;
real h=2+2*sqrt(7);
currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2));
currentlight=light(4,-4,4);
draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0));
draw(shift((1,3,1))*unitsphere,gray(0.85));
draw(shift((3,3,1))*unitsphere,gray(0.85));
draw(shift((3,1,1))*unitsphere,gray(0.85));
draw(shift((1,1,1))*unitsphere,gray(0.85));
draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85));
draw(shift((1,3,h-1))*unitsphere,gray(0.85));
draw(shift((3,3,h-1))*unitsphere,gray(0.85));
draw(shift((3,1,h-1))*unitsphere,gray(0.85));
draw(shift((1,1,h-1))*unitsphere,gray(0.85));
draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h));
[/asy]
$\textbf{(A) }2+2\sqrt 7\qquad
\textbf{(B) }3+2\sqrt 5\qquad
\textbf{(C) }4+2\sqrt 7\qquad
\textbf{(D) }4\sqrt 5\qquad
\textbf{(E) }4\sqrt 7\qquad$
1992 AMC 12/AHSME, 3
An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of the objects in the urn are gold coins?
$ \textbf{(A)}\ 40\%\qquad\textbf{(B)}\ 48\%\qquad\textbf{(C)}\ 52\%\qquad\textbf{(D)}\ 60\%\qquad\textbf{(E)}\ 80\% $
2020 AMC 12/AHSME, 11
A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square$?$
$\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{5}{8} \qquad \textbf{(C) } \frac{2}{3} \qquad \textbf{(D) } \frac{3}{4} \qquad \textbf{(E) } \frac{7}{8}$
1963 AMC 12/AHSME, 17
The expression $\dfrac{\dfrac{a}{a+y}+\dfrac{y}{a-y}}{\dfrac{y}{a+y}-\dfrac{a}{a-y}}$, a real, $a\neq 0$, has the value $-1$ for:
$\textbf{(A)}\ \text{all but two real values of }y \qquad
\textbf{(B)}\ \text{only two real values of }y \qquad$
$\textbf{(C)}\ \text{all real values of }y \qquad
\textbf{(D)}\ \text{only one real value of }y \qquad
\textbf{(E)}\ \text{no real values of }y$
2021 AMC 10 Spring, 10
An inverted cone with base radius $12 \text{ cm}$ and height $18 \text{ cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24 \text{ cm}$. What is the height in centimeters of the water in the cylinder?
$\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6$
2014 AMC 8, 16
The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?
$\textbf{(A) }60\qquad\textbf{(B) }88\qquad\textbf{(C) }96\qquad\textbf{(D) }144\qquad \textbf{(E) }160$
1988 AMC 12/AHSME, 26
Suppose that $p$ and $q$ are positive numbers for which \[ \log_{9}(p) = \log_{12}(q) = \log_{16}(p+q) \] What is the value of $\frac{q}{p}$?
$\textbf{(A)}\ \frac{4}{3}\qquad\textbf{(B)}\ \frac{1+\sqrt{3}}{2}\qquad\textbf{(C)}\ \frac{8}{5}\qquad\textbf{(D)}\ \frac{1+\sqrt{5}}{2}\qquad\textbf{(E)}\ \frac{16}{9} $
1986 AMC 12/AHSME, 23
Let \[N = 69^{5} + 5\cdot 69^{4} + 10\cdot 69^{3} + 10\cdot 69^{2} + 5\cdot 69 + 1.\] How many positive integers are factors of $N$?
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 125\qquad\textbf{(E)}\ 216 $
2024 AMC 12/AHSME, 10
Let $\alpha$ be the radian measure of the smallest angle in a $3{-}4{-}5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7{-}24{-}25$ right triangle. In terms of $\alpha$, what is $\beta$?
$
\textbf{(A) }\frac{\alpha}{3}\qquad
\textbf{(B) }\alpha - \frac{\pi}{8}\qquad
\textbf{(C) }\frac{\pi}{2} - 2\alpha \qquad
\textbf{(D) }\frac{\alpha}{2}\qquad
\textbf{(E) }\pi - 4\alpha\qquad
$
1981 USAMO, 3
If $A,B,C$ are the angles of a triangle, prove that
\[-2 \le \sin{3A}+\sin{3B}+\sin{3C} \le \frac{3\sqrt{3}}{2}\]
and determine when equality holds.
2024 AMC 12/AHSME, 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
$
2019 AMC 10, 7
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 $x+y=10 ?$
$\textbf{(A) } 4 \qquad\textbf{(B) } 4\sqrt{2} \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 6\sqrt{2}$
2021 AMC 10 Spring, 10
Which of the following is equivalent to $$(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$$
$\textbf{(A) }3^{127}+2^{127} \qquad \textbf{(B) }3^{127}+2^{127}+2\cdot 3^{63}+3\cdot 2^{63} \qquad \textbf{(C) }3^{128}-2^{128} \qquad \textbf{(D) }3^{128}+2^{128} \qquad \textbf{(E) }5^{127}$
2013 AMC 8, 21
Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 18$
2012 AMC 12/AHSME, 13
Two parabolas have equations $y=x^2+ax+b$ and $y=x^2+cx+d$, where $a$, $b$, $c$, and $d$ are integers (not necessarily different), each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common?
$\textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{25}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\ \frac{31}{36} \qquad\textbf{(E)}\ 1 $
2014 AMC 10, 11
For the consumer, a single discount of $n\%$ is more advantageous than any of the following discounts:
$(1)$ two successive $15\%$ discounts
$(2)$ three successive $10\%$ discounts
$(3)$ a $25\%$ discount followed by a $5\%$ discount
What is the smallest possible positive integer value of $n$?
${ \textbf{(A)}\ \ 27\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 29\qquad\textbf{(D)}}\ 31\qquad\textbf{(E)}\ 33$
1998 AMC 12/AHSME, 27
A $ 9\times9\times9$ cube is composed of twenty-seven $ 3\times3\times3$ cubes. The big cube is 'tunneled' as follows: First, the six $ 3\times3\times3$ cubes which make up the center of each face as well as the center of $ 3\times3\times3$ cube are removed. Second, each of the twenty remaining $ 3\times3\times3$ cubes is diminished in the same way. That is, the central facial unit cubes as well as each center cube are removed.
[asy]
import three;
size(4.5cm);
triple eye = (6, 9, 5);
currentprojection = perspective(eye);
real eps = 0.001;
for(int i = 0; i < 3; ++i){
for(int j = 0; j < 3; ++j){
for(int k = 0; k < 3; ++k){
if(i == 1 && j == 1) continue;
if(j == 1 && k == 1) continue;
if(k == 1 && i == 1) continue;
draw(shift(i, j, k) * scale(1 - eps, 1 - eps, 1 - eps) * unitcube, gray(0.9), nolight);
draw(shift(i, j, k) * (X--(X + Y)--Y--(Y+Z)--Z--(Z + X)--cycle));
draw(shift(i, j, k) * (X + Y + Z--X + Y));
draw(shift(i, j, k) * (X + Y + Z--Y + Z));
draw(shift(i, j, k) * (X + Y + Z--Z + X));
}
}
}
[/asy]
The surface area of the final figure is
$ \textbf{(A)}\ 384\qquad
\textbf{(B)}\ 729\qquad
\textbf{(C)}\ 864\qquad
\textbf{(D)}\ 1024\qquad
\textbf{(E)}\ 1056$
1978 AMC 12/AHSME, 2
If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is
$\textbf{(A) }\frac{1}{\pi^2}\qquad\textbf{(B) }\frac{1}{\pi}\qquad\textbf{(C) }1\qquad\textbf{(D) }\pi\qquad \textbf{(E) }\pi^2$
2014 AMC 12/AHSME, 18
The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is [i]bad[/i] if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?
$ \textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5 $