Found problems: 3632
2009 AMC 12/AHSME, 7
The first three terms of an arithmetic sequence are $ 2x\minus{}3$, $ 5x\minus{}11$, and $ 3x\plus{}1$ respectively. The $ n$th term of the sequence is $ 2009$. What is $ n$?
$ \textbf{(A)}\ 255 \qquad
\textbf{(B)}\ 502 \qquad
\textbf{(C)}\ 1004 \qquad
\textbf{(D)}\ 1506 \qquad
\textbf{(E)}\ 8037$
1979 AMC 12/AHSME, 29
For each positive number $x$, let \[f(x)=\displaystyle\frac{\left(x+\dfrac{1}{x}\right)^6-\left(x^6+\dfrac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}.\] The minimum value of $f(x)$ is
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }6$
2003 AMC 8, 21
The area of trapezoid $ ABCD$ is $ 164 \text{cm}^2$. The altitude is $ 8 \text{cm}$, $ AB$ is $ 10 \text{cm}$, and $ CD$ is $ 17 \text{cm}$. What is $ BC$, in centimeters?
[asy]/* AMC8 2003 #21 Problem */
size(4inch,2inch);
draw((0,0)--(31,0)--(16,8)--(6,8)--cycle);
draw((11,8)--(11,0), linetype("8 4"));
draw((11,1)--(12,1)--(12,0));
label("$A$", (0,0), SW);
label("$D$", (31,0), SE);
label("$B$", (6,8), NW);
label("$C$", (16,8), NE);
label("10", (3,5), W);
label("8", (11,4), E);
label("17", (22.5,5), E);[/asy]
$ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$
2006 AMC 12/AHSME, 20
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?
$ \textbf{(A) } \frac {1}{2187} \qquad \textbf{(B) } \frac {1}{729} \qquad \textbf{(C) } \frac {2}{243} \qquad \textbf{(D) } \frac {1}{81} \qquad \textbf{(E) } \frac {5}{243}$
1990 AMC 12/AHSME, 8
The number of real solutions of the equation \[|x-2|+|x-3|=1\] is
$\textbf{(A) }0\qquad
\textbf{(B) }1\qquad
\textbf{(C) }2\qquad
\textbf{(D) }3\qquad
\textbf{(E) }\text{more than }3\qquad$
1998 AMC 12/AHSME, 29
A point $ (x,y)$ in the plane is called a lattice point if both $ x$ and $ y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to
$ \textbf{(A)}\ 4.0\qquad
\textbf{(B)}\ 4.2\qquad
\textbf{(C)}\ 4.5\qquad
\textbf{(D)}\ 5.0\qquad
\textbf{(E)}\ 5.6$
2006 AMC 10, 13
Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
$ \textbf{(A) } \frac 67 \qquad \textbf{(B) } \frac {13}{14} \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \frac {14}{13} \qquad \textbf{(E) } \frac 76$
2020 AMC 10, 20
Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\frac{bc}{ad}?$
$\textbf{(A) } 6 \qquad\textbf{(B) } 19 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 26 \qquad\textbf{(E) } 38$
2017 AIME Problems, 14
A $10\times 10\times 10$ grid of points consists of all points in space of the form $(i,j,k)$, where $i$, $j$, and $k$ are integers between $1$ and $10$, inclusive. Find the number of different lines that contain exactly $8$ of these points.
1964 AMC 12/AHSME, 13
A circle is inscribed in a triangle with side lengths $8$, $13$, and $17$. Let the segments of the side of length $8$, made by a point of tangency, be $r$ and $s$, with $r<s$. What is the ratio $r:s$?
${{ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 2:5 \qquad\textbf{(C)}\ 1:2 \qquad\textbf{(D)}\ 2:3 }\qquad\textbf{(E)}\ 3:4 } $
2008 AIME Problems, 7
Let $ S_i$ be the set of all integers $ n$ such that $ 100i\leq n < 100(i \plus{} 1)$. For example, $ S_4$ is the set $ {400,401,402,\ldots,499}$. How many of the sets $ S_0, S_1, S_2, \ldots, S_{999}$ do not contain a perfect square?
2001 AMC 12/AHSME, 11
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
$ \displaystyle \textbf{(A)} \ \frac {3}{10} \qquad \textbf{(B)} \ \frac {2}{5} \qquad \textbf{(C)} \ \frac {1}{2} \qquad \textbf{(D)} \ \frac {3}{5} \qquad \textbf{(E)} \ \frac {7}{10}$
2009 AMC 12/AHSME, 3
Twenty percent less than $ 60$ is one-third more than what number?
$ \textbf{(A)}\ 16\qquad
\textbf{(B)}\ 30\qquad
\textbf{(C)}\ 32\qquad
\textbf{(D)}\ 36\qquad
\textbf{(E)}\ 48$
2014 AMC 12/AHSME, 9
Convex quadrilateral $ABCD$ has $AB = 3, BC = 4, CD = 13, AD = 12,$ and $\angle ABC = 90^\circ,$ as shown. What is the area of the quadrilateral?
[asy]
unitsize(.4cm);
defaultpen(linewidth(.8pt)+fontsize(14pt));
dotfactor=2;
pair A,B,C,D;
C = (0,0);
B = (0,4);
A = (3,4);
D = (12.8,-2.8);
draw(C--B--A--D--cycle);
draw(rightanglemark(C,B,A,20));
dot("$A$",A,N);
dot("$B$",B,NW);
dot("$C$",C,SW);
dot("$D$",D,E);
[/asy]
$ \textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 36 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 48 \qquad
\textbf{(E)}\ 58.5 $
1959 AMC 12/AHSME, 21
If $p$ is the perimeter of an equilateral triangle inscribed in a circle, the area of the circle is:
$ \textbf{(A)}\ \frac{\pi p^2}{3} \qquad\textbf{(B)}\ \frac{\pi p^2}{9}\qquad\textbf{(C)}\ \frac{\pi p^2}{27}\qquad\textbf{(D)}\ \frac{\pi p^2}{81} \qquad\textbf{(E)}\ \frac{\pi p^2 \sqrt3}{27} $
2022 AMC 10, 14
What is the number of ways the numbers from $1$ to $14$ can be split into $7$ pairs such that for each pair, the greater number is at least $2$ times the smaller number?
$\textbf{(A) }108\qquad\textbf{(B) }120\qquad\textbf{(C) }126\qquad\textbf{(D) }132\qquad\textbf{(E) }144$
1994 AMC 12/AHSME, 11
Three cubes of volume $1, 8$ and $27$ are glued together at their faces. The smallest possible surface area of the resulting configuration is
$ \textbf{(A)}\ 36 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 70 \qquad\textbf{(D)}\ 72 \qquad\textbf{(E)}\ 74 $
1969 AMC 12/AHSME, 23
For any integer $n$ greater than $1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is:
$\textbf{(A) }0\qquad
\textbf{(B) }1\qquad
\textbf{(C) }\dfrac n2\text{ for }n\text{ even,}\,\dfrac{n+1}2\text{ for }n\text{ odd}$
$\textbf{(D) }n-1\qquad
\textbf{(E) }n$
2016 AIME Problems, 8
For a permutation $p = (a_1,a_2,\ldots,a_9)$ of the digits $1,2,\ldots,9$, let $s(p)$ denote the sum of the three $3$-digit numbers $a_1a_2a_3$, $a_4a_5a_6$, and $a_7a_8a_9$. Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$. Let $n$ denote the number of permutations $p$ with $s(p) = m$. Find $|m - n|$.
2009 AMC 10, 19
A particular $ 12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $ 1$, it mistakenly displays a $ 9$. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?
$ \textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac58\qquad \textbf{(C)}\ \frac34\qquad \textbf{(D)}\ \frac56\qquad \textbf{(E)}\ \frac {9}{10}$
2013 AMC 10, 8
Ray's car averages 40 miles per gallon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?
$ \textbf{(A) }10\qquad\textbf{(B) }16\qquad\textbf{(C) }25\qquad\textbf{(D) }30\qquad\textbf{(E) }40 $
2008 AMC 10, 22
Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be $ 6$. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts $ 1$. If it comes up tails, he takes half of the previous term and subtracts $ 1$. What is the probability that the fourth term in Jacob's sequence is an integer?
$ \textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{5}{8} \qquad
\textbf{(E)}\ \frac{3}{4}$
2019 AMC 10, 22
Raashan, Sylvia, and Ted play the following game. Each starts with $\$1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $\$1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $\$1$? (For example, Raashan and Ted may each decide to give $\$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $\$0$, Sylvia would have $\$2$, and Ted would have $\$1$, and and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their $\$1$ to, and and the holdings will be the same as the end of the second [sic] round.
$\textbf{(A) } \frac{1}{7} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{2}{3}$
1980 AMC 12/AHSME, 11
If the sum of the first 10 terms and the sum of the first 100 terms of a given arithmetic progression are 100 and 10, respectively, then the sum of first 110 terms is:
$\text{(A)} \ 90 \qquad \text{(B)} \ -90 \qquad \text{(C)} \ 110 \qquad \text{(D)} \ -110 \qquad \text{(E)} \ -100$
2006 AMC 12/AHSME, 6
The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$?
[asy] unitsize(2mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,0)--(12,0)--(12,-4)); label("$D$",(0,4),NW); label("$C$",(18,4),NE); label("$B$",(18,-4),SE); label("$A$",(0,-4),SW); label("$y$",(9,1)); [/asy]$ \textbf{(A) } 6\qquad \textbf{(B) } 7\qquad \textbf{(C) } 8\qquad \textbf{(D) } 9\qquad \textbf{(E) } 10$