Found problems: 3632
2018 AMC 10, 6
A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required?
$\textbf{(A)} \frac{1}{15} \qquad \textbf{(B)} \frac{1}{10} \qquad \textbf{(C)} \frac{1}{6} \qquad \textbf{(D)} \frac{1}{5} \qquad \textbf{(E)} \frac{1}{4}$
2013 AMC 12/AHSME, 15
The number $2013$ is expressed in the form \[2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},\] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $
2012 Hanoi Open Mathematics Competitions, 12
In an isosceles triangle ABC with the base AB
given a point M $\in$ BC: Let O be the center of its circumscribed
circle and S be the center of the inscribed circle in ABC and
SM // AC: Prove that OM perpendicular BS.
2016 AMC 10, 13
Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
$\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$
2016 AMC 12/AHSME, 5
The War of $1812$ started with a declaration of war on Thursday, June $18$, $1812$. The peace treaty to end the war was signed $919$ days later, on December $24$, $1814$. On what day of the week was the treaty signed?
$\textbf{(A)}\ \text{Friday} \qquad
\textbf{(B)}\ \text{Saturday} \qquad
\textbf{(C)}\ \text{Sunday} \qquad
\textbf{(D)}\ \text{Monday} \qquad
\textbf{(E)}\ \text{Tuesday} $
1968 AMC 12/AHSME, 25
Ace runs with constant speed and Flash runs $x$ times as fast, $x>1$. Flash gives Ace a head start of $y$ yards, and, at a given signal, they start off in the same direction. Then the number of yards Flash must run to catch Ace is:
$\textbf{(A)}\ xy \qquad\textbf{(B)}\ \frac{y}{x+y} \qquad\textbf{(C)}\ \frac{xy}{x-1} \qquad\textbf{(D)}\ \frac{x+y}{x+1} \qquad\textbf{(E)}\ \frac{x+y}{x-1}$
2023 AMC 12/AHSME, 8
How many nonempty subsets $B$ of $\{0, 1, 2, 3, \dots, 12\}$ have the property that the number of elements in $B$ is equal to the least element of $B$? For example, $B = \{4, 6, 8, 11\}$ satisfies the condition.
$\textbf{(A)}\ 256 \qquad\textbf{(B)}\ 136 \qquad\textbf{(C)}\ 108 \qquad\textbf{(D)}\ 144 \qquad\textbf{(E)}\ 156$
2016 AMC 10, 18
Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
$\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$
1968 AMC 12/AHSME, 18
Side $AB$ of triangle $ABC$ has length $8$ inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is:
$\textbf{(A)}\ \dfrac{51}{4} \qquad
\textbf{(B)}\ 13 \qquad
\textbf{(C)}\ \dfrac{53}{4} \qquad
\textbf{(D)}\ \dfrac{40}{3} \qquad
\textbf{(E)}\ \dfrac{27}{2} $
1996 AIME Problems, 3
Find the smallest positive integer $n$ for which the expansion of $(xy - 3x +7y - 21)^n,$ after like terms have been collected, has at least 1996 terms.
2012 Math Prize For Girls Problems, 8
Suppose that $x$, $y$, and $z$ are real numbers such that $x + y + z = 3$ and $x^2 + y^2 + z^2 = 6$. What is the largest possible value of $z$?
2020 AMC 10, 11
What is the median of the following list of $4040$ numbers$?$
$$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$$
$\textbf{(A) } 1974.5 \qquad \textbf{(B) } 1975.5 \qquad \textbf{(C) } 1976.5 \qquad \textbf{(D) } 1977.5 \qquad \textbf{(E) } 1978.5$
2022 AMC 10, 8
A data set consists of $6$ (not distinct) positive integers: $1$, $7$, $5$, $2$, $5$, and $X$. The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all positive values of $X$?
$\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40$
2023 AMC 10, 17
A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$?
$\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$
2025 USAMO, 3
Alice the architect and Bob the builder play a game. First, Alice chooses two points $P$ and $Q$ in the plane and a subset $\mathcal{S}$ of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair $A,\,B$ of cities, they are connected with a road along the line segment $AB$ if and only if the following condition holds:
[center]For every city $C$ distinct from $A$ and $B$, there exists $R\in\mathcal{S}$ such[/center]
[center]that $\triangle PQR$ is directly similar to either $\triangle ABC$ or $\triangle BAC$.[/center]
Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy.
[i]Note:[/i] $\triangle UVW$ is [i]directly similar[/i] to $\triangle XYZ$ if there exists a sequence of rotations, translations, and dilations sending $U$ to $X$, $V$ to $Y$, and $W$ to $Z$.
2024 AIME, 8
Eight circles of radius $34$ can be placed tangent to side $\overline{BC}$ of $\triangle ABC$ such that the first circle is tangent to $\overline{AB}$, subsequent circles are externally tangent to each other, and the last is tangent to $\overline{AC}$. Similarly, $2024$ circles of radius $1$ can also be placed along $\overline{BC}$ in this manner. The inradius of $\triangle ABC$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2023 AMC 12/AHSME, 3
How many positive perfect squares less than $2023$ are divisible by $5$?
$\textbf{(A) } 8 \qquad\textbf{(B) }9 \qquad\textbf{(C) }10 \qquad\textbf{(D) }11 \qquad\textbf{(E) } 12$
1959 AMC 12/AHSME, 50
A club with $x$ members is organized into four committees in accordance with these two rules:
$ \text{(1)}\ \text{Each member belongs to two and only two committees}\qquad$
$\text{(2)}\ \text{Each pair of committees has one and only one member in common}$
Then $x$:
$\textbf{(A)} \ \text{cannont be determined} \qquad$
$\textbf{(B)} \ \text{has a single value between 8 and 16} \qquad$
$\textbf{(C)} \ \text{has two values between 8 and 16} \qquad$
$\textbf{(D)} \ \text{has a single value between 4 and 8} \qquad$
$\textbf{(E)} \ \text{has two values between 4 and 8} \qquad$
2022 AIME Problems, 8
Equilateral triangle $\triangle ABC$ is inscribed in circle $\omega$ with radius $18.$ Circle $\omega_A$ is tangent to sides $\overline{AB}$ and $\overline{AC}$ and is internally tangent to $\omega$. Circles $\omega_B$ and $\omega_C$ are defined analogously. Circles $\omega_A$, $\omega_B$, and $\omega_C$ meet in six points$-$two points for each pair of circles. The three intersection points closest to the vertices of $\triangle ABC$ are the vertices of a large equilateral triangle in the interior of $\triangle ABC$, and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of $\triangle ABC$. The side length of the smaller equilateral triangle can be written as $\sqrt{a}-\sqrt{b}$, where $a$ and $b$ are positive integers. Find $a+b$.
2009 AMC 10, 23
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every $ 90$ seconds, and Robert runs clockwise and completes a lap every $ 80$ seconds. Both start from the start line at the same time. At some random time between $ 10$ minutes and $ 11$ minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?
$ \textbf{(A)}\ \frac{1}{16}\qquad
\textbf{(B)}\ \frac18\qquad
\textbf{(C)}\ \frac{3}{16} \qquad
\textbf{(D)}\ \frac14\qquad
\textbf{(E)}\ \frac{5}{16}$
1959 AMC 12/AHSME, 45
If $\left(\log_3 x\right)\left(\log_x 2x\right)\left( \log_{2x} y\right)=\log_{x}x^2$, then $y$ equals:
$ \textbf{(A)}\ \frac92\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 81 $
2017 AMC 10, 12
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
2011 AMC 10, 7
Which of the following equations does NOT have a solution?
$\textbf{ (A) }\:(x+7)^2=0$ $\textbf{(B) }\:|-3x|+5=0$ $\textbf{ (C) }\:\sqrt{-x}-2=0$ $\textbf{ (D) }\:\sqrt{x}-8=0$ $\textbf{ (E) }\:|-3x|-4=0 $
2022 AMC 10, 12
A pair of fair $6$-sided dice is rolled $n$ times. What is the least value of $n$ such that the probability that the sum of the numbers face up on a roll equals $7$ at least once is greater than $\frac{1}{2}$?
$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6$
2007 AMC 10, 15
Four circles of radius $ 1$ are each tangent to two sides of a square and externally tangent to a circle of radius $ 2$, as shown. What is the area of the square?
[asy]unitsize(5mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
real h=3*sqrt(2)/2;
pair O0=(0,0), O1=(h,h), O2=(-h,h), O3=(-h,-h), O4=(h,-h);
pair X=O0+2*dir(30), Y=O2+dir(45);
draw((-h-1,-h-1)--(-h-1,h+1)--(h+1,h+1)--(h+1,-h-1)--cycle);
draw(Circle(O0,2));
draw(Circle(O1,1));
draw(Circle(O2,1));
draw(Circle(O3,1));
draw(Circle(O4,1));
draw(O0--X);
draw(O2--Y);
label("$2$",midpoint(O0--X),NW);
label("$1$",midpoint(O2--Y),SE);[/asy]$ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 22 \plus{} 12\sqrt {2}\qquad \textbf{(C)}\ 16 \plus{} 16\sqrt {3}\qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 36 \plus{} 16\sqrt {2}$