Found problems: 133
2021 AMC 12/AHSME Fall, 23
What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{ 1, 2, 3, …, 30\}$? (For example the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.)
$\textbf{(A)}\ \frac{2}{3} \qquad\textbf{(B)}\ \frac{29}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\
\frac{29}{30} \qquad\textbf{(E)}\ 1$
2020 AMC 10, 16
Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?
$\textbf{(A) } \text{Bela will always win.}$
$\textbf{(B) } \text{Jenn will always win.} $
$\textbf{(C) } \text{Bela will win if and only if }n \text{ is odd.}$
$\textbf{(D) } \text{Jenn will win if and only if }n \text{ is odd.} $
$\textbf{(E) } \text{Jenn will win if and only if }n > 8.$
2017 AMC 12/AHSME, 5
The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2=40$, first quartile $Q_1=33$, and third quartile $Q_3=43$. An outlier in a data set is a value that is more than $1.5$ times the interquartile range below the first quartile ($Q_1$) or more than $1.5$ times the interquartile range above the third quartile ($Q_3$), where the interquartile range is defined as $Q_3-Q_1$. How many outliers does this data set have?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$
2021 AMC 12/AHSME Fall, 15
Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c?$
[asy]
size(160);
defaultpen(linewidth(1.1));
path square = (1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle;
filldraw(square,white);
filldraw(rotate(30)*square,white);
filldraw(rotate(60)*square,white);
dot((0,0),linewidth(7));
[/asy]
$\textbf{(A)}\: 75\qquad\textbf{(B)} \: 93\qquad\textbf{(C)} \: 96\qquad\textbf{(D)} \: 129\qquad\textbf{(E)} \: 147$
2021 AMC 12/AHSME Fall, 19
Regular polygons with $5, 6, 7, $ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?
$\textbf{(A)}\ 52 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\
64 \qquad\textbf{(E)}\ 68$
2016 AMC 10, 18
In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$
2019 AMC 12/AHSME, 18
Square pyramid $ABCDE$ has base $ABCD,$ which measures $3$ cm on a side, and altitude $\overline{AE}$ perpendicular to the base$,$ which measures $6$ cm. Point $P$ lies on $\overline{BE},$ one third of the way from $B$ to $E;$ point $Q$ lies on $\overline{DE},$ one third of the way from $D$ to $E;$ and point $R$ lies on $\overline{CE},$ two thirds of the way from $C$ to $E.$ What is the area, in square centimeters, of $\triangle PQR?$
$\textbf{(A) } \frac{3\sqrt2}{2} \qquad\textbf{(B) } \frac{3\sqrt3}{2} \qquad\textbf{(C) } 2\sqrt2 \qquad\textbf{(D) } 2\sqrt3 \qquad\textbf{(E) } 3\sqrt2$
2017 AMC 10, 4
Suppose that $x$ and $y$ are nonzero real numbers such that \[\frac{3x+y}{x-3y}= -2.\] What is the value of \[\frac{x+3y}{3x-y}?\]
$\textbf{(A) } {-3} \qquad \textbf{(B) } {-1} \qquad \textbf{(C) } 1 \qquad \textbf{(D) }2 \qquad \textbf{(E) } 3$
2021 AMC 12/AHSME Fall, 14
Suppose that $P(z), Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z)=R(z)$. What is the minimum possible value of $N$?
$\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 5$
2021 AMC 10 Fall, 24
A cube is constructed from $4$ white unit cubes and $4$ black unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\
10 \qquad\textbf{(E)}\ 11$
2019 AMC 12/AHSME, 23
How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s?
$\textbf{(A) }55\qquad\textbf{(B) }60\qquad\textbf{(C) }65\qquad\textbf{(D) }70\qquad\textbf{(E) }75$
2018 AMC 12/AHSME, 7
What is the value of
\[ \log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27? \]
$\textbf{(A) } 3 \qquad \textbf{(B) } 3\log_{7}23 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 10 $
2021 AMC 12/AHSME Spring, 17
Let $ABCD$ be an isoceles trapezoid having parallel bases $\overline{AB}$ and $\overline{CD}$ with $AB>CD.$ Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $2, 3, 4,$ and $5$ starting with the triangle with base $\overline{CD}$ and moving clockwise as shown in the diagram below. What is the ratio $\frac{AB}{CD}?$[center][asy]unitsize(100);
pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5);
draw(A--B--C--D--cycle, black);
draw(A--P, black);
draw(B--P, black);
draw(C--P, black);
draw(D--P, black);
label("$A$",A,(-1,0));
label("$B$",B,(1,0));
label("$C$",C,(1,-0));
label("$D$",D,(-1,0));
label("$2$",E,(0,0));
label("$3$",F,(0,0));
label("$4$",G,(0,0));
label("$5$",H,(0,0));
dot(A^^B^^C^^D^^P);
[/asy][/center]
$\textbf{(A)}\: 3\qquad\textbf{(B)}\: 2+\sqrt{2}\qquad\textbf{(C)}\: 1+\sqrt{6}\qquad\textbf{(D)}\: 2\sqrt{3}\qquad\textbf{(E)}\: 3\sqrt{2}$
2020 AMC 12/AHSME, 23
How many integers $n \geq 2$ are there such that whenever $z_1, z_2, ..., z_n$ are complex numbers such that
$$|z_1| = |z_2| = ... = |z_n| = 1 \text{ and } z_1 + z_2 + ... + z_n = 0,$$
then the numbers $z_1, z_2, ..., z_n$ are equally spaced on the unit circle in the complex plane?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$
2019 AMC 12/AHSME, 10
The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L,$ without traveling along any portion of a road more than once. (Paula [i]is[/i] allowed to visit a city more than once.) How many different routes can Paula take?
[asy]
import olympiad;
unitsize(50);
for (int i = 0; i < 3; ++i) {
for (int j = 0; j < 4; ++j) {
pair A = (j,i);
dot(A);
}
}
for (int i = 0; i < 3; ++i) {
for (int j = 0; j < 4; ++j) {
if (j != 3) {
draw((j,i)--(j+1,i));
}
if (i != 2) {
draw((j,i)--(j,i+1));
}
}
}
label("$A$", (0,2), W);
label("$L$", (3,0), E);
[/asy]
$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$
2018 AMC 12/AHSME, 15
How many odd positive 3-digit integers are divisible by 3 but do not contain the digit 3?
$\textbf{(A) } 96 \qquad \textbf{(B) } 97 \qquad \textbf{(C) } 98 \qquad \textbf{(D) } 102 \qquad \textbf{(E) } 120 $
2016 AMC 10, 22
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A?$
$\textbf{(A)}\ 385 \qquad
\textbf{(B)}\ 665 \qquad
\textbf{(C)}\ 945 \qquad
\textbf{(D)}\ 1140 \qquad
\textbf{(E)}\ 1330$
2021 AMC 12/AHSME Spring, 19
Two fair dice, each with at least 6 faces, are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that die, inclusive. The probability of rolling a sum of 7 is $\frac{3}{4}$ of the probability of rolling a sum of 10 and the probability of rolling a sum of 12 is $\frac{1}{12}$. What is the least possible number of faces on the two dice combined?
$\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 19 \qquad\textbf{(E)}\ 20$
2018 AMC 12/AHSME, 3
A line with slope $2$ intersects a line with slope $6$ at the point $(40, 30)$. What is the distance between the $x$-intercepts of these two lines?
$\textbf{(A) }5\qquad\textbf{(B) }10\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }50$
2020 AMC 12/AHSME, 13
Which of the following is the value of $\sqrt{\log_2{6}+\log_3{6}}?$
$\textbf{(A) } 1 \qquad\textbf{(B) } \sqrt{\log_5{6}} \qquad\textbf{(C) } 2 \qquad\textbf{(D) } \sqrt{\log_2{3}}+\sqrt{\log_3{2}} \qquad\textbf{(E) } \sqrt{\log_2{6}}+\sqrt{\log_3{6}}$
2021 AMC 10 Fall, 21
Regular polygons with $5, 6, 7, $ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?
$\textbf{(A)}\ 52 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\
64 \qquad\textbf{(E)}\ 68$
2021 AMC 12/AHSME Fall, 8
The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle’s height to the base. What is the measure, in degrees, of the vertex angle of this triangle?
$\textbf{(A)}\ 105 \qquad\textbf{(B)}\ 120 \qquad\textbf{(C)}\ 135 \qquad\textbf{(D)}\
150 \qquad\textbf{(E)}\ 165$
2016 AMC 12/AHSME, 25
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n \ge 2$. What is the smallest positive integer $k$ such that the product $a_1a_2 \cdots a_k$ is an integer?
$\textbf{(A)}\ 17 \qquad
\textbf{(B)}\ 18 \qquad
\textbf{(C)}\ 19 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 21$
2020 AMC 12/AHSME, 6
For all integers $n \geq 9,$ the value of
$$\frac{(n+2)!-(n+1)!}{n!}$$
is always which of the following?
$\textbf{(A) } \text{a multiple of }4 \qquad \textbf{(B) } \text{a multiple of }10 \qquad \textbf{(C) } \text{a prime number} \\ \textbf{(D) } \text{a perfect square} \qquad \textbf{(E) } \text{a perfect cube}$
2016 AMC 12/AHSME, 21
Let $ABCD$ be a unit square. Let $Q_1$ be the midpoint of $\overline{CD}$. For $i=1,2,\dots,$ let $P_i$ be the intersection of $\overline{AQ_i}$ and $\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\overline{CD}$. What is
$$\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i \, ?$$
$\textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{1}{4} \qquad
\textbf{(C)}\ \frac{1}{3} \qquad
\textbf{(D)}\ \frac{1}{2} \qquad
\textbf{(E)}\ 1$