Found problems: 3632
2008 AIME Problems, 2
Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the $ 50$-mile mark at exactly the same time. How many minutes has it taken them?
2017 AMC 12/AHSME, 25
The vertices $V$ of a centrally symmetric hexagon in the complex plane are given by
$$V=\left\{ \sqrt{2}i,-\sqrt{2}i, \frac{1}{\sqrt{8}}(1+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}.$$
For each $j$, $1\leq j\leq 12$, an element $z_j$ is chosen from $V$ at random, independently of the other choices. Let $P={\prod}_{j=1}^{12}z_j$ be the product of the $12$ numbers selected. What is the probability that $P=-1$?
$\textbf{(A) } \dfrac{5\cdot11}{3^{10}} \qquad \textbf{(B) } \dfrac{5^2\cdot11}{2\cdot3^{10}} \qquad \textbf{(C) } \dfrac{5\cdot11}{3^{9}} \qquad \textbf{(D) } \dfrac{5\cdot7\cdot11}{2\cdot3^{10}} \qquad \textbf{(E) } \dfrac{2^2\cdot5\cdot11}{3^{10}}$
1987 AIME Problems, 7
Let $[r,s]$ denote the least common multiple of positive integers $r$ and $s$. Find the number of ordered triples $(a,b,c)$ of positive integers for which $[a,b] = 1000$, $[b,c] = 2000$, and $[c,a] = 2000$
1981 USAMO, 2
Every pair of communities in a county are linked directly by one mode of transportation; bus, train, or airplane. All three methods of transportation are used in the county with no community being serviced by all three modes and no three communities being linked pairwise by the same mode. Determine the largest number of communities in this county.
2019 AMC 12/AHSME, 14
For a certain complex number $c$, the polynomial
\[ P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\]
has exactly 4 distinct roots. What is $|c|$?
$\textbf{(A) } 2 \qquad \textbf{(B) } \sqrt{6} \qquad \textbf{(C) } 2\sqrt{2} \qquad \textbf{(D) } 3 \qquad \textbf{(E) } \sqrt{10}$
2009 AMC 12/AHSME, 12
The fifth and eighth terms of a geometric sequence of real numbers are $ 7!$ and $ 8!$ respectively. What is the first term?
$ \textbf{(A)}\ 60\qquad
\textbf{(B)}\ 75\qquad
\textbf{(C)}\ 120\qquad
\textbf{(D)}\ 225\qquad
\textbf{(E)}\ 315$
2014 AMC 10, 14
The $y$-intercepts, $P$ and $Q$, of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\triangle APQ$?
$ \textbf{(A)}\ 45\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 72 $
2022 AMC 12/AHSME, 25
Four regular hexagons surround a square with a side length $1$, each one sharing an edge with the square, as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be written as $m\sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime. What is $m + n + p$?
[asy]
import geometry;
unitsize(3cm);
draw((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle);
draw(shift((1/2,1-sqrt(3)/2))*polygon(6));
draw(shift((1/2,sqrt(3)/2))*polygon(6));
draw(shift((sqrt(3)/2,1/2))*rotate(90)*polygon(6));
draw(shift((1-sqrt(3)/2,1/2))*rotate(90)*polygon(6));
draw((0,1-sqrt(3))--(1,1-sqrt(3))--(3-sqrt(3),sqrt(3)-2)--(sqrt(3),0)--(sqrt(3),1)--(3-sqrt(3),3-sqrt(3))--(1,sqrt(3))--(0,sqrt(3))--(sqrt(3)-2,3-sqrt(3))--(1-sqrt(3),1)--(1-sqrt(3),0)--(sqrt(3)-2,sqrt(3)-2)--cycle,linewidth(2));
[/asy]
$\textbf{(A)}-12~\textbf{(B)}-4~\textbf{(C)} 4~\textbf{(D)}24~\textbf{(E)}32$
2009 AIME Problems, 4
A group of children held a grape-eating contest. When the contest was over, the winner had eaten $ n$ grapes, and the child in $ k$th place had eaten $ n\plus{}2\minus{}2k$ grapes. The total number of grapes eaten in the contest was $ 2009$. Find the smallest possible value of $ n$.
1964 AMC 12/AHSME, 40
A watch loses $2\frac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March 15. Let $n$ be the positive correction, in minutes, to be added to the time shown by the watch at a given time. When the watch shows $9$ A.M. on March 21, $n$ equals:
$\textbf{(A) }14\frac{14}{23}\qquad\textbf{(B) }14\frac{1}{14}\qquad\textbf{(C) }13\frac{101}{115}\qquad\textbf{(D) }13\frac{83}{115}\qquad \textbf{(E) }13\frac{13}{23}$
1972 AMC 12/AHSME, 22
If $a\pm bi~(b\neq 0)$ are imaginary roots of the equation $x^3+qx+r=0$ where $a,~b,~q,$ and $r$ are real numbers, then $q$ in terms of $a$ and $b$ is
$\textbf{(A) }a^2+b^2\qquad\textbf{(B) }2a^2-b^2\qquad\textbf{(C) }b^2-a^2\qquad\textbf{(D) }b^2-2a^2\qquad \textbf{(E) }b^2-3a^2$
1976 AMC 12/AHSME, 29
Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is $44$ years, then Ann's age is
$\textbf{(A) }22\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad \textbf{(E) }28$
2014 AMC 12/AHSME, 21
For every real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and let \[f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).\] The set of all numbers $x$ such that $1\leq x<2014$ and $f(x)\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals?
$\textbf{(A) }1\qquad
\textbf{(B) }\dfrac{\log 2015}{\log 2014}\qquad
\textbf{(C) }\dfrac{\log 2014}{\log 2013}\qquad
\textbf{(D) }\dfrac{2014}{2013}\qquad
\textbf{(E) }2014^{\frac1{2014}}\qquad$
1990 AMC 12/AHSME, 10
An $11\times 11\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?
$\textbf{(A) }328\qquad
\textbf{(B) }329\qquad
\textbf{(C) }330\qquad
\textbf{(D) }331\qquad
\textbf{(E) }332\qquad$
2024 AIME, 9
Let $A$, $B$, $C$, and $D$ be points in the coordinate plane on the hyperbola $\tfrac{x^{2}}{20}-\tfrac{y^{2}}{24}=1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^{2}$ for all such rhombi.
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]
$\textbf{(A)}\: 75\qquad\textbf{(B)} \: 93\qquad\textbf{(C)} \: 96\qquad\textbf{(D)} \: 129\qquad\textbf{(E)} \: 147$
2009 AMC 10, 17
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $ (a,0)$ to $ (3,3)$, divides the entire region into two regions of equal area. What is $ a$?
[asy]size(200);
defaultpen(linewidth(.8pt)+fontsize(8pt));
fill((2/3,0)--(3,3)--(3,1)--(2,1)--(2,0)--cycle,gray);
xaxis("$x$",-0.5,4,EndArrow(HookHead,4));
yaxis("$y$",-0.5,4,EndArrow(4));
draw((0,1)--(3,1)--(3,3)--(2,3)--(2,0));
draw((1,0)--(1,2)--(3,2));
draw((2/3,0)--(3,3));
label("$(a,0)$",(2/3,0),S);
label("$(3,3)$",(3,3),NE);[/asy]$ \textbf{(A)}\ \frac12\qquad
\textbf{(B)}\ \frac35\qquad
\textbf{(C)}\ \frac23\qquad
\textbf{(D)}\ \frac34\qquad
\textbf{(E)}\ \frac45$
2010 AIME Problems, 5
Positive numbers $ x$, $ y$, and $ z$ satisfy $ xyz \equal{} 10^{81}$ and $ (\log_{10}x)(\log_{10} yz) \plus{} (\log_{10}y) (\log_{10}z) \equal{} 468$. Find $ \sqrt {(\log_{10}x)^2 \plus{} (\log_{10}y)^2 \plus{} (\log_{10}z)^2}$.
2016 AIME Problems, 14
Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside of the plane of $\triangle ABC$ and are on the opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (The angle between the two planes). There is a point $O$ whose distance from each of $A,B,C,P$ and $Q$ is $d$. Find $d$.
1992 AMC 12/AHSME, 12
Let $y = mx + b$ be the image when the line $x - 3y + 11 = 0$ is reflected across the x-axis. The value of $m + b$ is
$ \textbf{(A)}\ -6\qquad\textbf{(B)}\ -5\qquad\textbf{(C)}\ -4\qquad\textbf{(D)}\ -3\qquad\textbf{(E)}\ -2 $
2013 USAMO, 4
Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]
2012 AMC 12/AHSME, 24
Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}...p_k^{e_k}$ is the prime factorization of $n>1$, then \[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.\] For every $m \ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$ in the range $1 \le N \le 400$ is the sequence $(f_1(N), f_2(N), f_3(N),...)$ unbounded?
[b]Note:[/b] a sequence of positive numbers is unbounded if for every integer $B$, there is a member of the sequence greater than $B$.
$ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 19 $
2021 AMC 12/AHSME Spring, 23
Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around'' and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up'', the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops?
$\textbf{(A) }\frac{9}{16}\qquad\textbf{(B) }\frac{5}{8}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{25}{32}\qquad\textbf{(E) }\frac{13}{16}$
2012 AMC 10, 21
Let points $A=(0,0,0)$, $B=(1,0,0)$, $C=(0,2,0)$, and $D=(0,0,3)$. Points $E,F,G$, and $H$ are midpoints of line segments $\overline{BD},\overline{AB},\overline{AC}$, and $\overline{DC}$ respectively. What is the area of $EFGH$?
$ \textbf{(A)}\ \sqrt2
\qquad\textbf{(B)}\ \frac{2\sqrt5}{3}
\qquad\textbf{(C)}\ \frac{3\sqrt5}{4}
\qquad\textbf{(D)}\ \sqrt3
\qquad\textbf{(E)}\ \frac{2\sqrt7}{3}
$
2007 AMC 12/AHSME, 2
An aquarium has a rectangular base that measures $ 100$ cm by $ 40$ cm and has a height of $ 50$ cm. It is filled with water to a height of $ 40$ cm. A brick with a rectangular base that measures $ 40$ cm by $ 20$ cm and a height of $ 10$ cm is placed in the aquarium. By how many centimeters does the water rise?
$ \textbf{(A)}\ 0.5 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 1.5 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 2.5$