Found problems: 57
2005 Romania National Olympiad, 3
Let $ABCD$ be a quadrilateral with $AB\parallel CD$ and $AC \perp BD$. Let $O$ be the intersection of $AC$ and $BD$. On the rays $(OA$ and $(OB$ we consider the points $M$ and $N$ respectively such that $\angle ANC = \angle BMD = 90^\circ$. We denote with $E$ the midpoint of the segment $MN$. Prove that
a) $\triangle OMN \sim \triangle OBA$;
b) $OE \perp AB$.
[i]Claudiu-Stefan Popa[/i]
2012 AMC 10, 15
In a round-robin tournament with $6$ teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 $
1982 AMC 12/AHSME, 16
A wooden cube has edges of length $3$ meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is
$\textbf {(A) } 54 \qquad \textbf {(B) } 72 \qquad \textbf {(C) } 76 \qquad \textbf {(D) } 84\qquad \textbf {(E) } 86$
1995 AMC 12/AHSME, 7
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is:
$\textbf{(A)}\ 25 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 75 \qquad
\textbf{(D)}\ 50 \qquad
\textbf{(E)}\ 100$
2014 Contests, 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]
2010 Contests, 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?.
2005 Harvard-MIT Mathematics Tournament, 5
A cube with side length $2$ is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?
2009 AIME Problems, 3
A coin that comes up heads with probability $ p > 0$ and tails with probability $ 1\minus{}p > 0$ independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to $ \frac{1}{25}$ of the probability of five heads and three tails. Let $ p \equal{} \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
2014 NIMO Problems, 2
How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube?
[i]Proposed by Evan Chen[/i]
2009 Puerto Rico Team Selection Test, 1
By the time a party is over, $ 28$ handshakes have occurred. If everyone shook everyone else's hand once, how many people attended the party?
2004 AMC 10, 19
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
[asy]
size(250);defaultpen(linewidth(0.8));
draw(ellipse(origin, 3, 1));
fill((3,0)--(3,2)--(-3,2)--(-3,0)--cycle, white);
draw((3,0)--(3,16)^^(-3,0)--(-3,16));
draw((0, 15)--(3, 12)^^(0, 16)--(3, 13));
filldraw(ellipse((0, 16), 3, 1), white, black);
draw((-3,11)--(3, 5)^^(-3,10)--(3, 4));
draw((-3,2)--(0,-1)^^(-3,1)--(-1,-0.89));
draw((0,-1)--(0,15), dashed);
draw((3,-2)--(3,-4)^^(-3,-2)--(-3,-4));
draw((-7,0)--(-5,0)^^(-7,16)--(-5,16));
draw((3,-3)--(-3,-3), Arrows(6));
draw((-6,0)--(-6,16), Arrows(6));
draw((-2,9)--(-1,9), Arrows(3));
label("$3$", (-1.375,9.05), dir(260), fontsize(7));
label("$A$", (0,15), N);
label("$B$", (0,-1), NE);
label("$30$", (0, -3), S);
label("$80$", (-6, 8), W);[/asy]
$ \textbf{(A)}\; 120\qquad
\textbf{(B)}\; 180\qquad
\textbf{(C)}\; 240\qquad
\textbf{(D)}\; 360\qquad
\textbf{(E)}\; 480$
2014 Purple Comet Problems, 13
Find $n>0$ such that $\sqrt[3]{\sqrt[3]{5\sqrt2+n}+\sqrt[3]{5\sqrt2-n}}=\sqrt2$.
2001 AMC 10, 2
A number $ x$ is $ 2$ more than the product of its reciprocal and its additive inverse. In which interval does the number lie?
$ \textbf{(A) }\minus{}4\le x\le \minus{}2\qquad\textbf{(B) }\minus{}2 < x\le 0\qquad\textbf{(C) }0 < x\le 2$
$ \textbf{(D) }2 < x\le 4\qquad\textbf{(E) }4 < x\le 6$
2008 AMC 10, 17
An equilateral triangle has side length $ 6$. What is the area of the region containing all points that are outside the triangle and not more than $ 3$ units from a point of the triangle?
$ \textbf{(A)}\ 36\plus{}24\sqrt{3} \qquad
\textbf{(B)}\ 54\plus{}9\pi \qquad
\textbf{(C)}\ 54\plus{}18\sqrt{3}\plus{}6\pi \qquad
\textbf{(D)}\ \left(2\sqrt{3}\plus{}3\right)^2\pi \\
\textbf{(E)}\ 9\left(\sqrt{3}\plus{}1\right)^2\pi$
1957 AMC 12/AHSME, 20
A man makes a trip by automobile at an average speed of $ 50$ mph. He returns over the same route at an average speed of $ 45$ mph. His average speed for the entire trip is:
$ \textbf{(A)}\ 47\frac{7}{19}\qquad
\textbf{(B)}\ 47\frac{1}{4}\qquad
\textbf{(C)}\ 47\frac{1}{2}\qquad
\textbf{(D)}\ 47\frac{11}{19}\qquad
\textbf{(E)}\ \text{none of these}$
2000 AMC 10, 8
At Olympic High School, $\frac25$ of the freshmen and $\frac45$ of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true?
$\text{(A)}$ There are five times as many sophomores as freshmen.
$\text{(B)}$ There are twice as many sophomores as freshmen.
$\text{(C)}$ There are as many freshmen as sophomores.
$\text{(D)}$ There are twice as many freshmen as sophomores.
$\text{(E)}$ There are five times as many freshmen as sophomores.
2013 AMC 12/AHSME, 19
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, 6
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$
2012 AMC 8, 6
A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inches high and 10 inches wide. What is the area of the border, in square inches?
$\textbf{(A)}\hspace{.05in}36 \qquad \textbf{(B)}\hspace{.05in}40 \qquad \textbf{(C)}\hspace{.05in}64 \qquad \textbf{(D)}\hspace{.05in}72 \qquad \textbf{(E)}\hspace{.05in}88 $
2014 NIMO Summer Contest, 2
How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube?
[i]Proposed by Evan Chen[/i]
1995 AIME Problems, 3
Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$
2014 Contests, 2
How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube?
[i]Proposed by Evan Chen[/i]
2011 Puerto Rico Team Selection Test, 2
How many 6-digit numbers have at least an even digit?
2010 AMC 10, 23
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$
2013 Purple Comet Problems, 9
Find the sum of all four-digit integers whose digits are a rearrangement of the digits $1$, $2$, $3$, $4$, such as $1234$, $1432$, or $3124$.