Found problems: 216
2014 NIMO Problems, 7
Let $\triangle ABC$ have $AB=6$, $BC=7$, and $CA=8$, and denote by $\omega$ its circumcircle. Let $N$ be a point on $\omega$ such that $AN$ is a diameter of $\omega$. Furthermore, let the tangent to $\omega$ at $A$ intersect $BC$ at $T$, and let the second intersection point of $NT$ with $\omega$ be $X$. The length of $\overline{AX}$ can be written in the form $\tfrac m{\sqrt n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Find $100m+n$.
[i]Proposed by David Altizio[/i]
1986 IMO Longlists, 76
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$
1966 AMC 12/AHSME, 6
$AB$ is the diameter of a circle centered at $O$. $C$ is a point on the circle such that angle $BOC$ is $60^\circ$. If the diameter of the circle is $5$ inches, the length of chord $AC$, expressed in inches, is:
$\text{(A)} \ 3 \qquad \text{(B)} \ \frac{5\sqrt{2}}{2} \qquad \text{(C)} \frac{5\sqrt3}{2} \ \qquad \text{(D)} \ 3\sqrt3 \qquad \text{(E)} \ \text{none of these}$
1990 Brazil National Olympiad, 3
Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$.
1998 Vietnam Team Selection Test, 2
In the plane we are given the circles $\Gamma$ and $\Delta$ tangent to each other and $\Gamma$ contains $\Delta$. The radius of $\Gamma$ is $R$ and of $\Delta$ is $\frac{R}{2}$. Prove that for each positive integer $n \geq 3$, the equation: \[ (p(1) - p(n))^2 = (n-1)^2 \cdot (2 \cdot (p(1) + p(n)) - (n-1)^2 - 8) \] is the necessary and sufficient condition for $n$ to exist $n$ distinct circles $\Upsilon_1, \Upsilon_2, \ldots, \Upsilon_n$ such that all these circles are tangent to $\Gamma$ and $\Delta$ and $\Upsilon_i$ is tangent to $\Upsilon_{i+1}$, and $\Upsilon_1$ has radius $\frac{R}{p(1)}$ and $\Upsilon_n$ has radius $\frac{R}{p(n)}$.
2013 Online Math Open Problems, 26
In triangle $ABC$, $F$ is on segment $AB$ such that $CF$ bisects $\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD,BE$ are perpendicular to $CF$. $M$ is the midpoint of $AB$. If $ME=13$, $AD=15$, and $BE=25$, find $AC+CB$.
[i]Ray Li[/i]
2006 Turkey Team Selection Test, 1
Find the maximum value for the area of a heptagon with all vertices on a circle and two diagonals perpendicular.
2011 International Zhautykov Olympiad, 1
Given is trapezoid $ABCD$, $M$ and $N$ being the midpoints of the bases of $AD$ and $BC$, respectively.
a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment $MN$.
b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line $MN$?
2010 Purple Comet Problems, 23
A disk with radius $10$ and a disk with radius $8$ are drawn so that the distance between their centers is $3$. Two congruent small circles lie in the intersection of the two disks so that they are tangent to each other and to each of the larger circles as shown. The radii of the smaller circles are both $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[asy]
size(150);
defaultpen(linewidth(1));
draw(circle(origin,10)^^circle((3,0),8)^^circle((5,15/4),15/4)^^circle((5,-15/4),15/4));
[/asy]
2005 China Team Selection Test, 2
Cyclic quadrilateral $ABCD$ has positive integer side lengths $AB$, $BC$, $CA$, $AD$. It is known that $AD=2005$, $\angle{ABC}=\angle{ADC} = 90^o$, and $\max \{ AB,BC,CD \} < 2005$. Determine the maximum and minimum possible values for the perimeter of $ABCD$.
2012 NIMO Problems, 5
In $\triangle ABC$, $AB = 30$, $BC = 40$, and $CA = 50$. Squares $A_1A_2BC$, $B_1B_2AC$, and $C_1C_2AB$ are erected outside $\triangle ABC$, and the pairwise intersections of lines $A_1A_2$, $B_1B_2$, and $C_1C_2$ are $P$, $Q$, and $R$. Compute the length of the shortest altitude of $\triangle PQR$.
[i]Proposed by Lewis Chen[/i]
2011 AMC 12/AHSME, 18
A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
$ \textbf{(A)}\ 5\sqrt{2}-7 \qquad
\textbf{(B)}\ 7-4\sqrt{3} \qquad
\textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad
\textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad
\textbf{(E)}\ \frac{\sqrt{3}}{9} $
2000 IMO Shortlist, 7
Ten gangsters are standing on a flat surface, and the distances between them are all distinct. At twelve o’clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed?
2011 Kosovo National Mathematical Olympiad, 4
In triangle $ABC$ medians of triangle $BE$ and $AD$ are perpendicular to each other. Find the length of $\overline{AB}$, if $\overline{BC}=6$ and $\overline{AC}=8$
1997 Turkey Junior National Olympiad, 2
Let $ABC$ be a triangle with $|AB|=|AC|=26$, $|BC|=20$. The altitudes of $\triangle ABC$ from $A$ and $B$ cut the opposite sides at $D$ and $E$, respectively. Calculate the radius of the circle passing through $D$ and tangent to $AC$ at $E$.
2007 China Team Selection Test, 2
Let $ I$ be the incenter of triangle $ ABC.$ Let $ M,N$ be the midpoints of $ AB,AC,$ respectively. Points $ D,E$ lie on $ AB,AC$ respectively such that $ BD\equal{}CE\equal{}BC.$ The line perpendicular to $ IM$ through $ D$ intersects the line perpendicular to $ IN$ through $ E$ at $ P.$ Prove that $ AP\perp BC.$
1998 AIME Problems, 10
Eight spheres of radius 100 are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of this last sphere is $a+b\sqrt{c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$
2014 AMC 12/AHSME, 21
In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?
[asy]
unitsize(150);
pair A,B,C,D,E,F,G,H,J,K;
A=(1,0); B=(0,0); C=(0,1); D=(1,1);
draw(A--B--C--D--A);
E=(2-sqrt(3),0); F=(2-sqrt(3),1);
draw(E--F);
G=(1,sqrt(3)/2); H=(2.5-sqrt(3),1);
K=(2-sqrt(3),1-sqrt(3)/2); J=(0.5,0);
draw(G--H--K--J--G);
label("$A$",A,SE);
label("$B$",B,SW);
label("$C$",C,NW);
label("$D$",D,NE);
label("$E$",E,S);
label("$F$",F,N);
label("$G$",G,E);
label("$H$",H,N);
label("$K$",K,W);
label("$J$",J,S);
[/asy]
$ \textbf{(A) }\dfrac{1}{2}(\sqrt{6}-2)\qquad\textbf{(B) }\dfrac{1}{4}\qquad\textbf{(C) }2-\sqrt{3}\qquad\textbf{(D) }\dfrac{\sqrt{3}}{6}\qquad\textbf{(E) }1-\dfrac{\sqrt{2}}{2} $
1984 IMO Longlists, 44
Let $a,b,c$ be positive numbers with $\sqrt{a}+\sqrt{b}+\sqrt{c}= \frac{\sqrt{3}}{2}$
Prove that the system of equations
\[\sqrt{y-a}+\sqrt{z-a}=1\]
\[\sqrt{z-b}+\sqrt{x-b}=1\]
\[\sqrt{x-c}+\sqrt{y-c}=1\]
has exactly one solution $(x,y,z)$ in real numbers.
It was proposed by Poland. Have fun! :lol:
2004 AMC 12/AHSME, 10
An [i]annulus[/i] is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \overline{OX}$ be a radius of the larger circle, let $ \overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \overline{OY}$ be the radius of the larger circle that contains $ Z$. Let $ a \equal{} XZ$, $ d \equal{} YZ$, and $ e \equal{} XY$. What is the area of the annulus?
$ \textbf{(A)}\ \pi a^2 \qquad \textbf{(B)}\ \pi b^2 \qquad \textbf{(C)}\ \pi c^2 \qquad \textbf{(D)}\ \pi d^2 \qquad \textbf{(E)}\ \pi e^2$
[asy]unitsize(1.4cm);
defaultpen(linewidth(.8pt));
dotfactor=3;
real r1=1.0, r2=1.8;
pair O=(0,0), Z=r1*dir(90), Y=r2*dir(90);
pair X=intersectionpoints(Z--(Z.x+100,Z.y), Circle(O,r2))[0];
pair[] points={X,O,Y,Z};
filldraw(Circle(O,r2),mediumgray,black);
filldraw(Circle(O,r1),white,black);
dot(points);
draw(X--Y--O--cycle--Z);
label("$O$",O,SSW,fontsize(10pt));
label("$Z$",Z,SW,fontsize(10pt));
label("$Y$",Y,N,fontsize(10pt));
label("$X$",X,NE,fontsize(10pt));
defaultpen(fontsize(8pt));
label("$c$",midpoint(O--Z),W);
label("$d$",midpoint(Z--Y),W);
label("$e$",midpoint(X--Y),NE);
label("$a$",midpoint(X--Z),N);
label("$b$",midpoint(O--X),SE);[/asy]
1981 AMC 12/AHSME, 2
Point $E$ is on side $AB$ of square $ABCD$. If $EB$ has length one and $EC$ has length two, then the area of the square is
$\text{(A)}\ \sqrt{3} \qquad \text{(B)}\ \sqrt{5} \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 2\sqrt{3} \qquad \text{(E)}\ 5$
2014 AIME Problems, 11
In $\triangle RED, RD =1, \angle DRE = 75^\circ$ and $\angle RED = 45^\circ$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC} \perp \overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA = AR$. Then $AE = \tfrac{a-\sqrt{b}}{c},$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer. Find $a+b+c$.
2014 Harvard-MIT Mathematics Tournament, 1
Let $O_1$ and $O_2$ be concentric circles with radii 4 and 6, respectively. A chord $AB$ is drawn in $O_1$ with length $2$. Extend $AB$ to intersect $O_2$ in points $C$ and $D$. Find $CD$.
2008 AMC 10, 10
Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$?
$ \textbf{(A)}\ \sqrt{10} \qquad
\textbf{(B)}\ \frac{7}{2} \qquad
\textbf{(C)}\ \sqrt{14} \qquad
\textbf{(D)}\ \sqrt{15} \qquad
\textbf{(E)}\ 4$
1993 AIME Problems, 15
Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime integers. Find $m + n$