Found problems: 329
2004 AMC 10, 20
In $ \triangle ABC$ points $ D$ and $ E$ lie on $ \overline{BC}$ and $ \overline{AC}$, respectively. If $ \overline{AD}$ and $ \overline{BE}$ intersect at $ T$ so that $ AT/DT \equal{} 3$ and $ BT/ET \equal{} 4$, what is $ CD/BD$?
[asy]unitsize(2cm);
defaultpen(linewidth(.8pt));
pair A = (0,0);
pair C = (2,0);
pair B = dir(57.5)*2;
pair E = waypoint(C--A,0.25);
pair D = waypoint(C--B,0.25);
pair T = intersectionpoint(D--A,E--B);
label("$B$",B,NW);label("$A$",A,SW);label("$C$",C,SE);label("$D$",D,NE);label("$E$",E,S);label("$T$",T,2*W+N);
draw(A--B--C--cycle);
draw(A--D);
draw(B--E);[/asy]$ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {2}{9}\qquad \textbf{(C)}\ \frac {3}{10}\qquad \textbf{(D)}\ \frac {4}{11}\qquad \textbf{(E)}\ \frac {5}{12}$
2004 Balkan MO, 4
The plane is partitioned into regions by a finite number of lines no three of which are concurrent. Two regions are called "neighbors" if the intersection of their boundaries is a segment, or half-line or a line (a point is not a segment). An integer is to be assigned to each region in such a way that:
i) the product of the integers assigned to any two neighbors is less than their sum;
ii) for each of the given lines, and each of the half-planes determined by it, the sum of the integers, assigned to all of the regions lying on this half-plane equal to zero.
Prove that this is possible if and only if not all of the lines are parallel.
2012 Online Math Open Problems, 42
In triangle $ABC,$ $\sin \angle A=\frac{4}{5}$ and $\angle A<90^\circ$ Let $D$ be a point outside triangle $ABC$ such that $\angle BAD=\angle DAC$ and $\angle BDC = 90^{\circ}.$ Suppose that $AD=1$ and that $\frac{BD} {CD} = \frac{3}{2}.$ If $AB+AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are pairwise relatively prime integers, find $a+b+c$.
[i]Author: Ray Li[/i]
2009 Princeton University Math Competition, 1
Find the number of pairs of integers $x$ and $y$ such that $x^2 + xy + y^2 = 28$.
1998 All-Russian Olympiad, 2
Two polygons are given on the plane. Assume that the distance between any two vertices of the same polygon is at most 1, and that the distance between any two vertices of different polygons is at least $ 1/\sqrt{2}$. Prove that these two polygons have no common interior points.
By the way, can two sides of a polygon intersect?
2012 APMO, 5
Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then
\[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \]
must hold.
2001 AMC 10, 25
How many positive integers not exceeding $ 2001$ are multiples of $ 3$ or $ 4$ but not $ 5$?
$ \textbf{(A)}\ 768 \qquad
\textbf{(B)}\ 801 \qquad
\textbf{(C)}\ 934 \qquad
\textbf{(D)}\ 1067 \qquad
\textbf{(E)}\ 1167$
2004 Korea - Final Round, 2
Prove that the equation $3y^2 = x^4 + x$ has no positive integer solutions.
1984 Putnam, A3
Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$ and let $M_n$ denote the $2n x 2n $ matrix whose $(i,j)$ entry $m_{ij}$ is given by
$m_{ij}=x$ if $i=j$,
$m_{ij}=a$ if $i \not= j$ and $i+j$ is even,
$m_{ij}=b$ if $i \not= j$ and $i+j$ is odd.
For example
$ M_2=\begin{vmatrix}x& b& a & b\\ b& x & b &a\\ a
& b& x & b\\ b & a & b & x \end{vmatrix}$.
Express $\lim_{x\to\ 0} \frac{ det M_n}{ (x-a)^{(2n-2)} }$ as a polynomial in $a,b $ and $n$ .
P.S. How write in latex $m_{ij}=...$ with symbol for the system (because is multiform function?)
1951 AMC 12/AHSME, 23
The radius of a cylindrical box is $ 8$ inches and the height is $ 3$ inches. The number of inches that may be added to either the radius or the height to give the same nonzero increase in volume is:
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 5\frac {1}{3} \qquad\textbf{(C)}\ \text{any number} \qquad\textbf{(D)}\ \text{non \minus{} existent} \qquad\textbf{(E)}\ \text{none of these}$
2000 Putnam, 5
Let $S_0$ be a finite set of positive integers. We define finite sets $S_1, S_2, \cdots$ of positive integers as follows: the integer $a$ in $S_{n+1}$ if and only if exactly one of $a-1$ or $a$ is in $S_n$. Show that there exist infinitely many integers $N$ for which $S_N = S_0 \cup \{ N + a: a \in S_0 \}$.
2007 AMC 10, 16
Integers $ a$, $ b$, $ c$, and $ d$, not necessarily distinct, are chosen independently and at random from $ 0$ to $ 2007$, inclusive. What is the probability that $ ad \minus{} bc$ is even?
$ \textbf{(A)}\ \frac {3}{8}\qquad \textbf{(B)}\ \frac {7}{16}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {9}{16}\qquad \textbf{(E)}\ \frac {5}{8}$
1963 AMC 12/AHSME, 36
A person starting with $64$ cents and making $6$ bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is:
$\textbf{(A)}\ \text{a loss of } 27 \qquad
\textbf{(B)}\ \text{a gain of }27 \qquad
\textbf{(C)}\ \text{a loss of }37 \qquad$
$
\textbf{(D)}\ \text{neither a gain nor a loss} \qquad
\textbf{(E)}\ \text{a gain or a loss depending upon the order in which the wins and losses occur}$
Note: Due to the lack of $\LaTeX$ packages, the numbers in the answer choices are in cents ¢
2003 AMC 12-AHSME, 15
A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.
[asy]unitsize(2.5cm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
filldraw(Circle((0,.866),.5),grey,black);
label("1",(0,.866),S);
filldraw(Circle((0,0),1),white,black);
draw((-.5,.866)--(.5,.866),linetype("4 4"));
clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle);
draw((-1,0)--(1,0));
label("2",(0,0),S);[/asy]$ \textbf{(A)}\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad \textbf{(B)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad \textbf{(C)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad\textbf{(D)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi$
$ \textbf{(E)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi$
2014 AMC 10, 22
In rectangle $ABCD$, $AB=20$ and $BC=10$. Let $E$ be a point on $\overline{CD}$ such that $\angle CBE=15^\circ$. What is $AE$?
$ \textbf{(A)}\ \dfrac{20\sqrt3}3\qquad\textbf{(B)}\ 10\sqrt3\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 11\sqrt3\qquad\textbf{(E)}\ 20 $
2006 AMC 12/AHSME, 16
Regular hexagon $ ABCDEF$ has vertices $ A$ and $ C$ at $ (0,0)$ and $ (7,1)$, respectively. What is its area?
$ \textbf{(A) } 20\sqrt {3} \qquad \textbf{(B) } 22\sqrt {3} \qquad \textbf{(C) } 25\sqrt {3} \qquad \textbf{(D) } 27\sqrt {3} \qquad \textbf{(E) } 50$
2004 Romania Team Selection Test, 1
Let $a_1,a_2,a_3,a_4$ be the sides of an arbitrary quadrilateral of perimeter $2s$. Prove that
\[ \sum\limits^4_{i=1} \dfrac 1{a_i+s} \leq \dfrac 29\sum\limits_{1\leq i<j\leq 4} \dfrac 1{ \sqrt { (s-a_i)(s-a_j)}}. \]
When does the equality hold?
2012 AIME Problems, 12
Let $\triangle ABC$ be a right triangle with right angle at $C$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{8}{15}$, then $\tan B$ can be written as $\frac{m\sqrt{p}}{n}$, where $m$ and $n$ are relatively prime positive integers, and $p$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.
1999 Flanders Math Olympiad, 1
Determine all 6-digit numbers $(abcdef)$ so that $(abcdef) = (def)^2$ where $\left( x_1x_2...x_n \right)$ is no multiplication but an n-digit number.
1975 AMC 12/AHSME, 30
Let $x=\cos 36^{\circ} - \cos 72^{\circ}$. Then $x$ equals
$ \textbf{(A)}\ \frac{1}{3} \qquad\textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ 3-\sqrt{6} \qquad\textbf{(D)}\ 2\sqrt{3}-3 \qquad\textbf{(E)}\ \text{none of these} $
2003 AMC 10, 19
A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.
[asy]unitsize(2.5cm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
filldraw(Circle((0,.866),.5),grey,black);
label("1",(0,.866),S);
filldraw(Circle((0,0),1),white,black);
draw((-.5,.866)--(.5,.866),linetype("4 4"));
clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle);
draw((-1,0)--(1,0));
label("2",(0,0),S);[/asy]$ \textbf{(A)}\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad \textbf{(B)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad \textbf{(C)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad\textbf{(D)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi$
$ \textbf{(E)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi$
1998 AMC 8, 24
A rectangular board of 8 columns has squared numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result?
[asy]
unitsize(20);
for(int a = 0; a < 10; ++a)
{
draw((0,a)--(8,a));
}
for (int b = 0; b < 9; ++b)
{
draw((b,0)--(b,9));
}
draw((0,0)--(0,-.5));
draw((1,0)--(1,-1.5));
draw((.5,-1)--(1.5,-1));
draw((2,0)--(2,-.5));
draw((4,0)--(4,-.5));
draw((5,0)--(5,-1.5));
draw((4.5,-1)--(5.5,-1));
draw((6,0)--(6,-.5));
draw((8,0)--(8,-.5));
fill((0,8)--(1,8)--(1,9)--(0,9)--cycle,black);
fill((2,8)--(3,8)--(3,9)--(2,9)--cycle,black);
fill((5,8)--(6,8)--(6,9)--(5,9)--cycle,black);
fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,black);
fill((6,7)--(7,7)--(7,8)--(6,8)--cycle,black);
label("$2$",(1.5,8.2),N);
label("$4$",(3.5,8.2),N);
label("$5$",(4.5,8.2),N);
label("$7$",(6.5,8.2),N);
label("$8$",(7.5,8.2),N);
label("$9$",(0.5,7.2),N);
label("$11$",(2.5,7.2),N);
label("$12$",(3.5,7.2),N);
label("$13$",(4.5,7.2),N);
label("$14$",(5.5,7.2),N);
label("$16$",(7.5,7.2),N);
[/asy]
$\text{(A)}\ 36 \qquad \text{(B)}\ 64 \qquad \text{(C)}\ 78 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 120$
2004 China Team Selection Test, 1
Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$
2007 Turkey Team Selection Test, 3
We write $1$ or $-1$ on each unit square of a $2007 \times 2007$ board. Find the number of writings such that for every square on the board the absolute value of the sum of numbers on the square is less then or equal to $1$.
2004 China Team Selection Test, 2
Let $p_1, p_2, \ldots, p_{25}$ are primes which don’t exceed 2004. Find the largest integer $T$ such that every positive integer $\leq T$ can be expressed as sums of distinct divisors of $(p_1\cdot p_2 \cdot \ldots \cdot p_{25})^{2004}.$