Found problems: 3632
2016 AMC 12/AHSME, 22
For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\frac{1}{n}$ is $0.\overline{abcdef}$, a repeating decimal of period $6$, and the decimal equivalent of $\frac{1}{n+6}$ is $0.\overline{wxyz}$, a repeating decimal of period $4$. In which interval does $n$ lie?
$\textbf{(A)}\ [1,200] \qquad
\textbf{(B)}\ [201,400] \qquad
\textbf{(C)}\ [401,600] \qquad
\textbf{(D)}\ [601,800] \qquad
\textbf{(E)}\ [801,999] $
2006 AIME Problems, 1
In quadrilateral $ABCD, \angle B$ is a right angle, diagonal $\overline{AC}$ is perpendicular to $\overline{CD},$ $AB=18, BC=21,$ and $CD=14.$ Find the perimeter of $ABCD$.
2012 AMC 8, 25
A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$ ?
[asy]
draw((0,2)--(2,2)--(2,0)--(0,0)--cycle);
draw((0,0.3)--(0.3,2)--(2,1.7)--(1.7,0)--cycle);
label("$a$",(-0.1,0.15));
label("$b$",(-0.1,1.15));
[/asy]
$\textbf{(A)}\hspace{.05in}\dfrac15 \qquad \textbf{(B)}\hspace{.05in}\dfrac25 \qquad \textbf{(C)}\hspace{.05in}\dfrac12 \qquad \textbf{(D)}\hspace{.05in}1 \qquad \textbf{(E)}\hspace{.05in}4 $
2009 AIME Problems, 6
How many positive integers $ N$ less than $ 1000$ are there such that the equation $ x^{\lfloor x\rfloor} \equal{} N$ has a solution for $ x$? (The notation $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$.)
1985 AIME Problems, 8
The sum of the following seven numbers is exactly 19:
\[a_1=2.56,\qquad a_2=2.61,\qquad a_3=2.65,\qquad a_4=2.71,\]
\[a_5=2.79,\qquad a_6=2.82,\qquad a_7=2.86.\]
It is desired to replace each $a_i$ by an integer approximation $A_i$, $1 \le i \le 7$, so that the sum of the $A_i$'s is also 19 and so that $M$, the maximum of the "errors" $|A_i - a_i|$, is as small as possible. For this minimum $M$, what is $100M$?
2008 AMC 10, 20
Trapezoid $ ABCD$ has bases $ \overline{AB}$ and $ \overline{CD}$ and diagonals intersecting at $ K$. Suppose that $ AB\equal{}9$, $ DC\equal{}12$, and the area of $ \triangle AKD$ is $ 24$. What is the area of trapezoid $ ABCD$?
$ \textbf{(A)}\ 92 \qquad
\textbf{(B)}\ 94 \qquad
\textbf{(C)}\ 96 \qquad
\textbf{(D)}\ 98 \qquad
\textbf{(E)}\ 100$
1979 AMC 12/AHSME, 23
The edges of a regular tetrahedron with vertices $A ,~ B,~ C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$.
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{4}\qquad\textbf{(C) }\frac{\sqrt{2}}{2}\qquad\textbf{(D) }\frac{\sqrt{3}}{2}\qquad\textbf{(E) }\frac{\sqrt{3}}{3}$
[asy]
size(150);
import patterns;
pair D=(0,0),C=(1,-1),B=(2.5,-0.2),A=(1,2),AA,BB,CC,DD,P,Q,aux;
add("hatch",hatch());
//AA=new A and etc.
draw(rotate(100,D)*(A--B--C--D--cycle));
AA=rotate(100,D)*A;
BB=rotate(100,D)*D;
CC=rotate(100,D)*C;
DD=rotate(100,D)*B;
aux=midpoint(AA--BB);
draw(BB--DD);
P=midpoint(AA--aux);
aux=midpoint(CC--DD);
Q=midpoint(CC--aux);
draw(AA--CC,dashed);
dot(P);
dot(Q);
fill(DD--BB--CC--cycle,pattern("hatch"));
label("$A$",AA,W);
label("$B$",BB,S);
label("$C$",CC,E);
label("$D$",DD,N);
label("$P$",P,S);
label("$Q$",Q,E);
//Credit to TheMaskedMagician for the diagram
[/asy]
2023 AMC 8, 5
A lake contains $250$ trout, along with a variety of other fish. When a marine biologist catches and releases a sample of $180$ fish from the lake, $30$ are identified as trout. Assume that the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake?
$\textbf{(A)}~1250\qquad \textbf{(B)}~1500\qquad \textbf{(C)}~1750\qquad \textbf{(D)}~1800\qquad \textbf{(E)}~2000$
2002 AIME Problems, 5
Find the sum of all positive integers $a=2^{n}3^{m},$ where $n$ and $m$ are non-negative integers, for which $a^{6}$ is not a divisor of $6^{a}.$
1968 AMC 12/AHSME, 10
Assume that, for a certain school, it is true that
[list]I: Some students are not honest
II: All fraternity members are honest[/list]
A necessary conclusion is:
$\textbf{(A)}\ \text{Some students are fraternity members} \qquad\\
\textbf{(B)}\ \text{Some fraternity members are not students} \qquad\\
\textbf{(C)}\ \text{Some students are not fraternity members} \qquad\\
\textbf{(D)}\ \text{No fraternity member is a student} \qquad\\
\textbf{(E)}\ \text{No student is a fraternity member} $
1996 AMC 8, 6
What is the smallest result that can be obtained from the following process?
*Choose three different numbers from the set $\{3,5,7,11,13,17\}$.
*Add two of these numbers.
*Multiply their sum by the third number.
$\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 50 \qquad \text{(E)}\ 56$
1969 AMC 12/AHSME, 7
If the points $(1,y_1)$ and $(-1,y_2)$ lie on the graph of $y=ax^2+bx+c$, and $y_1-y_2=-6$, then $b$ equals:
$\textbf{(A) }-3\qquad
\textbf{(B) }0\qquad
\textbf{(C) }3\qquad
\textbf{(D) }\sqrt{ac}\qquad
\textbf{(E) }\dfrac{a+c}2$
2014 AMC 12/AHSME, 25
What is the sum of all positive real solutions $x$ to the equation \[2\cos(2x)\left(\cos(2x)-\cos\left(\frac{2014\pi^2}{x}\right)\right)=\cos(4x)-1?\]
$\textbf{(A) }\pi\qquad
\textbf{(B) }810\pi\qquad
\textbf{(C) }1008\pi\qquad
\textbf{(D) }1080\pi\qquad
\textbf{(E) }1800\pi\qquad$
1961 AMC 12/AHSME, 4
Let the set consisting of the squares of the positive integers be called $u$; thus $u$ is the set $1, 4, 9, 16 . . .$. If a certain operation on one or more members of the set always yields a member of the set, we say that the set is closed under that operation. Then $u$ is closed under:
${{ \textbf{(A)}\ \text{Addition}\qquad\textbf{(B)}\ \text{Multiplication} \qquad\textbf{(C)}\ \text{Division} \qquad\textbf{(D)}\ \text{Extraction of a positive integral root} }\qquad\textbf{(E)} \text{None of these} } $
2018 AMC 12/AHSME, 13
Square $ABCD$ has side length $30$. Point $P$ lies inside the square so that $AP = 12$ and $BP = 26$. The centroids of $\triangle{ABP}$, $\triangle{BCP}$, $\triangle{CDP}$, and $\triangle{DAP}$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral?
[asy]
unitsize(120);
pair B = (0, 0), A = (0, 1), D = (1, 1), C = (1, 0), P = (1/4, 2/3);
draw(A--B--C--D--cycle);
dot(P);
defaultpen(fontsize(10pt));
draw(A--P--B);
draw(C--P--D);
label("$A$", A, W);
label("$B$", B, W);
label("$C$", C, E);
label("$D$", D, E);
label("$P$", P, N*1.5+E*0.5);
dot(A);
dot(B);
dot(C);
dot(D);
[/asy]
$\textbf{(A) }100\sqrt{2}\qquad\textbf{(B) }100\sqrt{3}\qquad\textbf{(C) }200\qquad\textbf{(D) }200\sqrt{2}\qquad\textbf{(E) }200\sqrt{3}$
1977 AMC 12/AHSME, 17
Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one?
$\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{1}{27}\qquad\textbf{(D) }\frac{1}{54}\qquad \textbf{(E) }\frac{7}{36}$
2006 AMC 12/AHSME, 13
Rhombus $ ABCD$ is similar to rhombus $ BFDE$. The area of rhombus $ ABCD$ is 24, and $ \angle BAD \equal{} 60^\circ$. What is the area of rhombus $ BFDE$?
[asy]
size(180);
defaultpen(linewidth(0.7)+fontsize(11));
pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3));
pair point=(3/2, sqrt(3)/2);
draw(B--C--D--A--B--F--D--E--B);
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));[/asy]
$ \textbf{(A) } 6 \qquad \textbf{(B) } 4\sqrt {3} \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 6\sqrt {3}$
2011 AMC 12/AHSME, 15
How many positive two-digit integers are factors of $2^{24} -1$?
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 10 \qquad
\textbf{(D)}\ 12 \qquad
\textbf{(E)}\ 14 $
2009 AMC 10, 2
Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could [i]not[/i] be the total value of the four coins, in cents?
$ \textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 25 \qquad
\textbf{(C)}\ 35 \qquad
\textbf{(D)}\ 45 \qquad
\textbf{(E)}\ 55$
2011 AIME Problems, 6
Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \le a<b<c<d \le 10$, and $a+d>b+c$. How many ordered quadruples are there?
2012 AMC 10, 5
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is $10\%$. She leaves a $15\%$ tip on the prices of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of $27.50$ for dinner. What is the cost of here dinner without tax or tip?
$ \textbf{(A)}\ \$18\qquad\textbf{(B)}\ \$20\qquad\textbf{(C)}\ \$21\qquad\textbf{(D)}\ \$22\qquad\textbf{(E)}\ \$24$
1979 AMC 12/AHSME, 13
The inequality $y-x<\sqrt{x^2}$ is satisfied if and only if
$\textbf{(A) }y<0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad$
$\textbf{(B) }y>0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad$
$\textbf{(C) }y^2<2xy\qquad\textbf{(D) }y<0\qquad\textbf{(E) }x>0\text{ and }y<2x$
2010 AMC 12/AHSME, 1
Makayla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35$
1996 AIME Problems, 9
A bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each closed locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens?
2010 AMC 12/AHSME, 20
Arithmetic sequences $ (a_n)$ and $ (b_n)$ have integer terms with $ a_1 \equal{} b_1 \equal{} 1 < a_2 \le b_2$ and $ a_nb_n \equal{} 2010$ for some $ n$. What is the largest possible value of $ n$?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 8 \qquad
\textbf{(D)}\ 288 \qquad
\textbf{(E)}\ 2009$