Found problems: 3632
1966 AMC 12/AHSME, 34
Let $r$ be the speed in miles per hour at which a wheel, $11$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\tfrac{1}{4}$ of a second, the speed $r$ is increased by $5$ miles per hour. The $r$ is:
$\text{(A)}\ 9\qquad
\text{(B)}\ 10\qquad
\text{(C)}\ 10\tfrac{1}{2}\qquad
\text{(D)}\ 11\qquad
\text{(E)}\ 12$
1972 AMC 12/AHSME, 10
For $x$ real, the inequality $1\le |x-2|\le 7$ is equivalent to
$\textbf{(A) }x\le 1\text{ or }x\ge 3\qquad\textbf{(B) }1\le x\le 3\qquad\textbf{(C) }-5\le x\le 9\qquad$
$\textbf{(D) }-5\le x\le 1\text{ or }3\le x\le 9\qquad \textbf{(E) }-6\le x\le 1\text{ or }3\le x\le 10$
1992 AMC 12/AHSME, 20
Part of an "$n$-pointed regular star" is shown. It is a simple closed polygon in which all $2n$ edges are congruent, angles $A_{1}$, $A_{2}$, $\ldots$, $A_{n}$ are congruent and angles $B_{1}$, $B_{2}$, $\ldots$, $B_{n}$ are congruent. If the acute angle at $A_{1}$ is $10^{\circ}$ less than the acute angle at $B_{1}$, then $n = $
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=dir(90-2*36), B=dir(90-36), C=dir(90), D=dir(90+36), E=dir(90+2*36);
pair F=2*dir(90-1.5*36), G=2*dir(90-0.5*36), H=2*dir(90+0.5*36), I=2*dir(90+1.5*36);
draw(A--F--B--G--C--H--D--I--E);
label("$B_2$", B, -0.3*dir(B));
label("$B_1$", C, -0.3*dir(C));
label("$B_n$", D, -0.3*dir(D));
label("$A_3$", F, dir(F));
label("$A_2$", G, dir(G));
label("$A_1$", H, dir(H));
label("$A_n$", I, dir(I));
[/asy]
$ \textbf{(A)}\ 12\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 36\qquad\textbf{(E)}\ 60 $
2024 AMC 12/AHSME, 23
What is the value of \[\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16}+\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16}+\tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?\]
$\textbf{(A) } 28 \qquad \textbf{(B) } 68 \qquad \textbf{(C) } 70 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 84$
1974 AMC 12/AHSME, 2
Let $x_1$ and $x_2$ be such that $x_1 \neq x_2$ and $3x_i^2-hx_i=b$, $i=1, 2$. Then $x_1+x_2$ equals
$ \textbf{(A)}\ -\frac{h}{3} \qquad\textbf{(B)}\ \frac{h}{3} \qquad\textbf{(C)}\ \frac{b}{3} \qquad\textbf{(D)}\ 2b \qquad\textbf{(E)}\ -\frac{b}{3} $
2012 AMC 10, 13
An [i]iterative average[/i] of the numbers $1$, $2$, $3$, $4$, and $5$ is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
$ \textbf{(A)}\ \frac{31}{16}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{17}{8}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \frac{65}{16} $
1993 AMC 12/AHSME, 22
Twenty cubical blocks are arranged as shown. First, $10$ are arranged in a triangular pattern; then a layer of $6$, arranged in a triangular pattern, is centered on the $10$; then a layer of $3$, arranged in a triangular pattern, is centered on the $6$; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered $1$ through $10$ in some order. Each block in layers $2, 3$ and $4$ is assigned the number which is the sum of the numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.
[asy]
size((400));
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0), linewidth(1));
draw((5,0)--(10,0)--(15,0)--(20,0)--(20,5)--(15,5)--(10,5)--(5,5)--(6,7)--(11,7)--(16,7)--(21,7)--(21,2)--(20,0), linewidth(1));
draw((10,0)--(10,5)--(11,7), linewidth(1));
draw((15,0)--(15,5)--(16,7), linewidth(1));
draw((20,0)--(20,5)--(21,7), linewidth(1));
draw((0,5)--(1,7)--(6,7), linewidth(1));
draw((3.5,7)--(4.5,9)--(9.5,9)--(14.5,9)--(19.5,9)--(18.5,7)--(19.5,9)--(19.5,7), linewidth(1));
draw((8.5,7)--(9.5,9), linewidth(1));
draw((13.5,7)--(14.5,9), linewidth(1));
draw((7,9)--(8,11)--(13,11)--(18,11)--(17,9)--(18,11)--(18,9), linewidth(1));
draw((12,9)--(13,11), linewidth(1));
draw((10.5,11)--(11.5,13)--(16.5,13)--(16.5,11)--(16.5,13)--(15.5,11), linewidth(1));
draw((25,0)--(30,0)--(30,5)--(25,5)--(25,0), dashed);
draw((30,0)--(35,0)--(40,0)--(45,0)--(45,5)--(40,5)--(35,5)--(30,5)--(31,7)--(36,7)--(41,7)--(46,7)--(46,2)--(45,0), dashed);
draw((35,0)--(35,5)--(36,7), dashed);
draw((40,0)--(40,5)--(41,7), dashed);
draw((45,0)--(45,5)--(46,7), dashed);
draw((25,5)--(26,7)--(31,7), dashed);
draw((28.5,7)--(29.5,9)--(34.5,9)--(39.5,9)--(44.5,9)--(43.5,7)--(44.5,9)--(44.5,7), dashed);
draw((33.5,7)--(34.5,9), dashed);
draw((38.5,7)--(39.5,9), dashed);
draw((32,9)--(33,11)--(38,11)--(43,11)--(42,9)--(43,11)--(43,9), dashed);
draw((37,9)--(38,11), dashed);
draw((35.5,11)--(36.5,13)--(41.5,13)--(41.5,11)--(41.5,13)--(40.5,11), dashed);
draw((50,0)--(55,0)--(55,5)--(50,5)--(50,0), dashed);
draw((55,0)--(60,0)--(65,0)--(70,0)--(70,5)--(65,5)--(60,5)--(55,5)--(56,7)--(61,7)--(66,7)--(71,7)--(71,2)--(70,0), dashed);
draw((60,0)--(60,5)--(61,7), dashed);
draw((65,0)--(65,5)--(66,7), dashed);
draw((70,0)--(70,5)--(71,7), dashed);
draw((50,5)--(51,7)--(56,7), dashed);
draw((53.5,7)--(54.5,9)--(59.5,9)--(64.5,9)--(69.5,9)--(68.5,7)--(69.5,9)--(69.5,7), dashed);
draw((58.5,7)--(59.5,9), dashed);
draw((63.5,7)--(64.5,9), dashed);
draw((57,9)--(58,11)--(63,11)--(68,11)--(67,9)--(68,11)--(68,9), dashed);
draw((62,9)--(63,11), dashed);
draw((60.5,11)--(61.5,13)--(66.5,13)--(66.5,11)--(66.5,13)--(65.5,11), dashed);
draw((75,0)--(80,0)--(80,5)--(75,5)--(75,0), dashed);
draw((80,0)--(85,0)--(90,0)--(95,0)--(95,5)--(90,5)--(85,5)--(80,5)--(81,7)--(86,7)--(91,7)--(96,7)--(96,2)--(95,0), dashed);
draw((85,0)--(85,5)--(86,7), dashed);
draw((90,0)--(90,5)--(91,7), dashed);
draw((95,0)--(95,5)--(96,7), dashed);
draw((75,5)--(76,7)--(81,7), dashed);
draw((78.5,7)--(79.5,9)--(84.5,9)--(89.5,9)--(94.5,9)--(93.5,7)--(94.5,9)--(94.5,7), dashed);
draw((83.5,7)--(84.5,9), dashed);
draw((88.5,7)--(89.5,9), dashed);
draw((82,9)--(83,11)--(88,11)--(93,11)--(92,9)--(93,11)--(93,9), dashed);
draw((87,9)--(88,11), dashed);
draw((85.5,11)--(86.5,13)--(91.5,13)--(91.5,11)--(91.5,13)--(90.5,11), dashed);
draw((28,6)--(33,6)--(38,6)--(43,6)--(43,11)--(38,11)--(33,11)--(28,11)--(28,6), linewidth(1));
draw((28,11)--(29,13)--(34,13)--(39,13)--(44,13)--(43,11)--(44,13)--(44,8)--(43,6), linewidth(1));
draw((33,6)--(33,11)--(34,13)--(39,13)--(38,11)--(38,6), linewidth(1));
draw((31,13)--(32,15)--(37,15)--(36,13)--(37,15)--(42,15)--(41,13)--(42,15)--(42,13), linewidth(1));
draw((34.5,15)--(35.5,17)--(40.5,17)--(39.5,15)--(40.5,17)--(40.5,15), linewidth(1));
draw((53,6)--(58,6)--(63,6)--(68,6)--(68,11)--(63,11)--(58,11)--(53,11)--(53,6), dashed);
draw((53,11)--(54,13)--(59,13)--(64,13)--(69,13)--(68,11)--(69,13)--(69,8)--(68,6), dashed);
draw((58,6)--(58,11)--(59,13)--(64,13)--(63,11)--(63,6), dashed);
draw((56,13)--(57,15)--(62,15)--(61,13)--(62,15)--(67,15)--(66,13)--(67,15)--(67,13), dashed);
draw((59.5,15)--(60.5,17)--(65.5,17)--(64.5,15)--(65.5,17)--(65.5,15), dashed);
draw((78,6)--(83,6)--(88,6)--(93,6)--(93,11)--(88,11)--(83,11)--(78,11)--(78,6), dashed);
draw((78,11)--(79,13)--(84,13)--(89,13)--(94,13)--(93,11)--(94,13)--(94,8)--(93,6), dashed);
draw((83,6)--(83,11)--(84,13)--(89,13)--(88,11)--(88,6), dashed);
draw((81,13)--(82,15)--(87,15)--(86,13)--(87,15)--(92,15)--(91,13)--(92,15)--(92,13), dashed);
draw((84.5,15)--(85.5,17)--(90.5,17)--(89.5,15)--(90.5,17)--(90.5,15), dashed);
draw((56,12)--(61,12)--(66,12)--(66,17)--(61,17)--(56,17)--(56,12), linewidth(1));
draw((61,12)--(61,17)--(62,19)--(57,19)--(56,17)--(57,19)--(67,19)--(66,17)--(67,19)--(67,14)--(66,12), linewidth(1));
draw((59.5,19)--(60.5,21)--(65.5,21)--(64.5,19)--(65.5,21)--(65.5,19), linewidth(1));
draw((81,12)--(86,12)--(91,12)--(91,17)--(86,17)--(81,17)--(81,12), dashed);
draw((86,12)--(86,17)--(87,19)--(82,19)--(81,17)--(82,19)--(92,19)--(91,17)--(92,19)--(92,14)--(91,12), dashed);
draw((84.5,19)--(85.5,21)--(90.5,21)--(89.5,19)--(90.5,21)--(90.5,19), dashed);
draw((84,18)--(89,18)--(89,23)--(84,23)--(84,18)--(84,23)--(85,25)--(90,25)--(89,23)--(90,25)--(90,20)--(89,18), linewidth(1));[/asy]
$ \textbf{(A)}\ 55 \qquad\textbf{(B)}\ 83 \qquad\textbf{(C)}\ 114 \qquad\textbf{(D)}\ 137 \qquad\textbf{(E)}\ 144 $
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?
$\textbf{(A)}\ 105 \qquad\textbf{(B)}\ 120 \qquad\textbf{(C)}\ 135 \qquad\textbf{(D)}\
150 \qquad\textbf{(E)}\ 165$
1999 AIME Problems, 14
Point $P$ is located inside traingle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=13, BC=14,$ and $CA=15,$ and the tangent of angle $PAB$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2018 AMC 10, 2
Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alica have?
$ \textbf{(A) }\text{ Liliane has } 20\%\text{ more soda than Alice.}$
$\textbf{(B) }\text{ Liliane has } 25\%\text{ more soda than Alice.}$
$\textbf{(C) }\text{ Liliane has } 45\%\text{ more soda than Alice.}$
$ \textbf{(D) }\text{ Liliane has } 75\%\text{ more soda than Alice.}$
$\textbf{(E) }\text{ Liliane has } 100\%\text{ more soda than Alice.}$
2024 AMC 10, 21
The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5{.}$ The numbers in positions $(5, 5), \,(2,4),\,(4,3),$ and $(3, 1)$ are $0, 48, 16,$ and $12{,}$ respectively. What number is in position $(1, 2)?$
\[ \begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}\]
$\textbf{(A) } 19 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 29 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 39$
1989 AMC 12/AHSME, 26
A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is
$ \textbf{(A)}\ \frac{\sqrt{3}}{12} \qquad\textbf{(B)}\ \frac{\sqrt{6}}{16} \qquad\textbf{(C)}\ \frac{1}{6} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{8} \qquad\textbf{(E)}\ \frac{1}{4} $
2010 USAJMO, 1
A [i]permutation[/i] of the set of positive integers $[n] = \{1, 2, . . . , n\}$ is a sequence $(a_1 , a_2 , \ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P (n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P (n)$ is a multiple of $2010$.
2024 USAJMO, 4
Let $n \geq 3$ be an integer. Rowan and Colin play a game on an $n \times n$ grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is [i]orderly[/i] if: [list] [*]no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and [*]no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring. [/list] In terms of $n$, how many orderly colorings are there?
[i]Proposed by Alec Sun[/i]
2022 AIME Problems, 6
Let $x_1\leq x_2\leq \cdots\leq x_{100}$ be real numbers such that $|x_1| + |x_2| + \cdots + |x_{100}| = 1$ and $x_1 + x_2 + \cdots + x_{100} = 0$. Among all such $100$-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2008 AMC 12/AHSME, 16
The numbers $ \log(a^3b^7)$, $ \log(a^5b^{12})$, and $ \log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $ 12^\text{th}$ term of the sequence is $ \log{b^n}$. What is $ n$?
$ \textbf{(A)}\ 40 \qquad
\textbf{(B)}\ 56 \qquad
\textbf{(C)}\ 76 \qquad
\textbf{(D)}\ 112 \qquad
\textbf{(E)}\ 143$
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?
$\textbf{(A)}\ 17 \qquad
\textbf{(B)}\ 18 \qquad
\textbf{(C)}\ 19 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 21$
2015 AMC 12/AHSME, 18
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$?
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
2010 AMC 10, 9
A [i]palindrome[/i], such as $ 83438$, is a number that remains the same when its digits are reversed. The numbers $ x$ and $ x \plus{} 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x?
$ \textbf{(A)}\ 20\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 23\qquad \textbf{(E)}\ 24$
1963 AMC 12/AHSME, 15
A circle is inscribed in an equilateral triangle, and a square is inscribed in the circle. The ratio of the area of the triangle to the area of the square is:
$\textbf{(A)}\ \sqrt{3}:1 \qquad
\textbf{(B)}\ \sqrt{3}:\sqrt{2} \qquad
\textbf{(C)}\ 3\sqrt{3}:2 \qquad
\textbf{(D)}\ 3:\sqrt{2} \qquad
\textbf{(E)}\ 3:2\sqrt{2}$
2022 AIME Problems, 9
Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$. Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i=1,2,3,\ldots, m$ and $j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$. The figure shows that there are 8 regions when $m=3$ and $n=2$.
[asy]
import geometry;
size(10cm);
draw((-2,0)--(13,0));
draw((0,4)--(10,4));
label("$\ell_A$",(-2,0),W);
label("$\ell_B$",(0,4),W);
point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2);
draw(B1--A1--B2);
draw(B1--A2--B2);
draw(B1--A3--B2);
label("$A_1$",A1,S);
label("$A_2$",A2,S);
label("$A_3$",A3,S);
label("$B_1$",B1,N);
label("$B_2$",B2,N);
label("1",centroid(A1,B1,I1));
label("2",centroid(B1,I1,I3));
label("3",centroid(B1,B2,I3));
label("4",centroid(A1,A2,I1));
label("5",(A2+I1+I2+I3)/4);
label("6",centroid(B2,I2,I3));
label("7",centroid(A2,A3,I2));
label("8",centroid(A3,B2,I2));
dot(A1);
dot(A2);
dot(A3);
dot(B1);
dot(B2);
[/asy]
2012 AIME Problems, 1
Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \neq 0$, $c \neq 0$ such that both $abc$ and $cba$ are divisible by 4.
2011 AMC 12/AHSME, 4
At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12, 15, and 10 minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?
$ \textbf{(A)}\ 12 \qquad
\textbf{(B)}\ \frac{37}{3} \qquad
\textbf{(C)}\ \frac{88}{7}\qquad
\textbf{(D)}\ 13 \qquad
\textbf{(E)}\ 14$
2017 AMC 10, 16
There are $10$ horses, named Horse 1, Horse 2, $\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$?
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$