Found problems: 57
2004 AMC 8, 20
Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are $6$ empty chairs, how many people are in the room?
$\textbf{(A)}\ 12\qquad
\textbf{(B)}\ 18\qquad
\textbf{(C)}\ 24\qquad
\textbf{(D)}\ 27\qquad
\textbf{(E)}\ 36$
1984 AIME Problems, 3
A point $P$ is chosen in the interior of $\triangle ABC$ so that when lines are drawn through $P$ parallel to the sides of $\triangle ABC$, the resulting smaller triangles, $t_1$, $t_2$, and $t_3$ in the figure, have areas 4, 9, and 49, respectively. Find the area of $\triangle ABC$.
[asy]
size(200);
pathpen=black+linewidth(0.65);pointpen=black;
pair A=(0,0),B=(12,0),C=(4,5);
D(A--B--C--cycle); D(A+(B-A)*3/4--A+(C-A)*3/4); D(B+(C-B)*5/6--B+(A-B)*5/6);D(C+(B-C)*5/12--C+(A-C)*5/12);
MP("A",C,N);MP("B",A,SW);MP("C",B,SE); /* sorry mixed up points according to resources diagram. */
MP("t_3",(A+B+(B-A)*3/4+(A-B)*5/6)/2+(-1,0.8),N);
MP("t_2",(B+C+(B-C)*5/12+(C-B)*5/6)/2+(-0.3,0.1),WSW);
MP("t_1",(A+C+(C-A)*3/4+(A-C)*5/12)/2+(0,0.15),ESE);[/asy]
2014 AMC 10, 19
Where is AMC 10a No.19? Thanks
2002 AMC 10, 2
The sum of eleven consecutive integers is $2002$. What is the smallest of these integers?
$\textbf{(A) }175\qquad\textbf{(B) }177\qquad\textbf{(C) }179\qquad\textbf{(D) }180\qquad\textbf{(E) }181$
2010 AMC 12/AHSME, 17
The entries in a $ 3\times3$ array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there?
$ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 42\qquad\textbf{(E)}\ 60$
2005 AMC 12/AHSME, 19
A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?
$ \textbf{(A)}\ 1404 \qquad \textbf{(B)}\ 1462 \qquad \textbf{(C)}\ 1604 \qquad \textbf{(D)}\ 1605 \qquad \textbf{(E)}\ 1804$
2013 AMC 10, 23
In $ \bigtriangleup ABC $, $ AB = 86 $, and $ AC = 97 $. A circle with center $ A $ and radius $ AB $ intersects $ \overline{BC} $ at points $ B $ and $ X $. Moreover $ \overline{BX} $ and $ \overline{CX} $ have integer lengths. What is $ BC $?
$ \textbf{(A)} \ 11 \qquad \textbf{(B)} \ 28 \qquad \textbf{(C)} \ 33 \qquad \textbf{(D)} \ 61 \qquad \textbf{(E)} \ 72 $
2010 AMC 12/AHSME, 18
A 16-step path is to go from $ ( \minus{} 4, \minus{}4)$ to $ (4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $ \minus{} 2 \le x \le 2$, $ \minus{} 2 \le y \le 2$ at each step?
$ \textbf{(A)}\ 92 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 1568 \qquad \textbf{(D)}\ 1698 \qquad \textbf{(E)}\ 12,\!800$
1990 AIME Problems, 1
The increasing sequence $2,3,5,6,7,10,11,\ldots$ consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence.
1972 AMC 12/AHSME, 1
The lengths in inches of the three sides of each of four triangles I, II, III, and IV are as follows:
\[ \begin{array}{rlrl} \hbox {I}& 3,\ 4,\ \hbox{and}\ 5 \qquad & \hbox{III}& 7,\ 24,\ \hbox{and}\ 25 \\ \hbox{II}& 4,\ 7\frac{1}{2},\ \hbox{and}\ 8\frac{1}{2} \qquad & \hbox{IV}& 3\frac{1}{2},\ 4\frac{1}{2},\ \hbox{and}\ 5\frac{1}{2}. \end{array} \]
Of these four given triangles, the only right triangles are
\[ \begin{tabular}{rlrlrl} (A) & I and II \qquad & (B) & I and III \qquad & (C) & I and IV \\ (D) & I, II, and III \qquad & (E) & I, II, and IV & \end{tabular} \]
1999 AMC 12/AHSME, 2
Which of the following statements is false?
$ \textbf{(A)}\ \text{All equilateral triangles are congruent to each other.}$
$ \textbf{(B)}\ \text{All equilateral triangles are convex.}$
$ \textbf{(C)}\ \text{All equilateral triangles are equilangular.}$
$ \textbf{(D)}\ \text{All equilateral triangles are regular polygons.}$
$ \textbf{(E)}\ \text{All equilateral triangles are similar to each other.}$
2003 AMC 8, 16
Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has $4$ seats: $1$ Driver seat, $1$ front passenger seat, and $2$ back passenger seat. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there?
$\textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 12 \qquad
\textbf{(E)}\ 24$
2001 AMC 12/AHSME, 2
Let $ P(n)$ and $ S(n)$ denote the product and the sum, respectively, of the digits of the integer $ n$. For example, $ P(23) \equal{} 6$ and $ S(23) \equal{} 5$. Suppose $ N$ is a two-digit number such that $ N \equal{} P(N) \plus{} S(N)$. What is the units digit of $ N$?
$ \textbf{(A)} \ 2 \qquad \textbf{(B)} \ 3 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 8 \qquad \textbf{(E)} \ 9$
2013 AMC 10, 1
What is $\frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}$?
$\textbf{(A) }-1\qquad\textbf{(B) }\frac5{36}\qquad\textbf{(C) }\frac7{12}\qquad\textbf{(D) }\frac{49}{20}\qquad\textbf{(E) }\frac{43}3$
2002 Moldova National Olympiad, 2
Can a square of side $ 1024$ be partitioned into $ 31$ squares?Can a square of side $ 1023$ be partitioned into $ 30$ squares, one of which has a s side lenght not exceeding $ 1$?
2010 Paraguay Mathematical Olympiad, 2
A series of figures is shown in the picture below, each one of them created by following a secret rule. If the leftmost figure is considered the first figure, how many squares will the 21st figure have?
[img]http://www.artofproblemsolving.com/Forum/download/file.php?id=49934[/img]
Note: only the little squares are to be counted (i.e., the $2 \times 2$ squares, $3 \times 3$ squares, $\dots$ should not be counted)
Extra (not part of the original problem): How many squares will the 21st figure have, if we consider all $1 \times 1$ squares, all $2 \times 2$ squares, all $3 \times 3$ squares, and so on?.
1972 AMC 12/AHSME, 5
From among $2^{1/2},$ $3^{1/3},$ $8^{1/8},$ $9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are
\[ \begin{array}{rlrlrlrl} \hbox {(A)}& 3^{1/3},\ 2^{1/2} \quad & \hbox {(B)}& 3^{1/3},\ 8^{1/8} \quad & \hbox {(C)}& 3^{1/3},\ 9^{1/9} \quad & \hbox {(D)}& 8^{1/8},\ 9^{1/9} \\ \hbox {(E)}& \multicolumn{3}{l}{\hbox{None of these}} \end{array} \]
1987 AIME Problems, 11
Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.
1992 AIME Problems, 1
Find the sum of all positive rational numbers that are less than $10$ and that have denominator $30$ when written in lowest terms.
2000 AIME Problems, 15
Find the least positive integer $n$ such that \[ \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}. \]
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$
2014 NIMO Problems, 3
A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square?
[i]Proposed by Evan Chen[/i]
1951 AMC 12/AHSME, 48
The area of a square inscribed in a semicircle is to the area of the square inscribed in the entire circle as:
$ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 2: 3 \qquad\textbf{(C)}\ 2: 5 \qquad\textbf{(D)}\ 3: 4 \qquad\textbf{(E)}\ 3: 5$
1972 AMC 12/AHSME, 32
[asy]
real t=pi/12;real u=8*t;
real cu=cos(u);real su=sin(u);
draw(unitcircle);
draw((cos(-t),sin(-t))--(cos(13*t),sin(13*t)));
draw((cu,su)--(cu,-su));
label("A",(cos(13*t),sin(13*t)),W);
label("B",(cos(-t),sin(-t)),E);
label("C",(cu,su),N);
label("D",(cu,-su),S);
label("E",(cu,sin(-t)),NE);
label("2",((cu-1)/2,sin(-t)),N);
label("6",((cu+1)/2,sin(-t)),N);
label("3",(cu,(sin(-t)-su)/2),E);
//Credit to Zimbalono for the diagram[/asy]
Chords $AB$ and $CD$ in the circle above intersect at $E$ and are perpendicular to each other. If segments $AE$, $EB$, and $ED$ have measures $2$, $3$, and $6$ respectively, then the length of the diameter of the circle is
$\textbf{(A) }4\sqrt{5}\qquad\textbf{(B) }\sqrt{65}\qquad\textbf{(C) }2\sqrt{17}\qquad\textbf{(D) }3\sqrt{7}\qquad \textbf{(E) }6\sqrt{2}$
2000 AMC 10, 18
Charlyn walks completely around the boundary of a square whose sides are each $5$ km long. From any point on her path she can see exactly $1$ km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?
$\text{(A)}\ 24 \qquad\text{(B)}\ 27\qquad\text{(C)}\ 39\qquad\text{(D)}\ 40 \qquad\text{(E)}\ 42$