Found problems: 970
In $ \triangle ABC$, we have $ AC \equal{} BC \equal{} 7$ and $ AB \equal{} 2$. Suppose that $ D$ is a point on line $ AB$ such that $ B$ lies between $ A$ and $ D$ and $ CD \equal{} 8$. What is $ BD$?
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 2 \sqrt {3}\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ 4 \sqrt {2}$
Let $\Omega_1$ and $\Omega_2$ be two circles in the plane. Suppose the common external tangent to $\Omega_1$ and $\Omega_2$ has length $2017$ while their common internal tangent has length $2009$. Find the product of the radii of $\Omega_1$ and $\Omega_2$.
[i]Proposed by David Altizio[/i]
In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?
$ \textbf{(A)}\ 12+9\sqrt{3}\qquad\textbf{(B)}\ 18+6\sqrt{3}\qquad\textbf{(C)}\ 12+12\sqrt{2}\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 32 $
Let $k$ be a positive integer. Show that there are infinitely many positive integer solutions $(m, n)$ to
$(m - n)^2 = kmn + m + n$.
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$.
[i]Proposed by C.R. Pranesachar, India [/i]
Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma $ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^{\circ}$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order.
Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.
Proposed by Ukraine
Suppose that every integer has been given one of the colours red, blue, green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$. Show that there are two integers of the same colour whose difference has one of the following values: $x,y,x+y$ or $x-y$.
A square of side length $ 1$ and a circle of radius $ \sqrt3/3$ share the same center. What is the area inside the circle, but outside the square?
$ \textbf{(A)}\ \frac{\pi}3 \minus{} 1 \qquad\textbf{(B)}\ \frac{2\pi}{9} \minus{} \frac{\sqrt3}3 \qquad\textbf{(C)}\ \frac{\pi}{18} \qquad\textbf{(D)}\ \frac14 \qquad\textbf{(E)}\ 2\pi/9$
Points $A(6,13)$ and $B(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$?
$\textbf{(A) }\frac{83\pi}{8}\qquad\textbf{(B) }\frac{21\pi}{2}\qquad\textbf{(C) }
\frac{85\pi}{8}\qquad\textbf{(D) }\frac{43\pi}{4}\qquad\textbf{(E) }\frac{87\pi}{8}$
In a triangle $ABC$ the area is $18$, the length $AB$ is $5$, and the medians from $A$ and $B$ are orthogonal. Find the lengths of the sides $BC,AC$.
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let the internal angle bisector of $\angle BAC$ intersect $BC$ and $\Gamma$ at $E$ and $N$, respectively. Let $A'$ be the antipode of $A$ on $\Gamma$ and let $V$ be the point where $AA'$ intersects $BC$. Given that $EV=6$, $VA'=7$, and $A'N=9$, compute the radius of $\Gamma$.
[i]Proposed by James Lin.[/i]
A right cone in $xyz$-space has its apex at $(0,0,0)$, and the endpoints of a diameter on its base are $(12,13,-9)$ and $(12,-5,15)$. The volume of the cone can be expressed as $a\pi$. What is $a$?
In a convex quadrilateral $PQRS$, $PQ =RS$, $(\sqrt{3} +1 )QR = SP$ and $\angle RSP - \angle SQP = 30^{\circ}$. Prove that $\angle PQR - \angle QRS = 90^{\circ}.$
Let $ABC$ be a triangle with $AB = AC$. Let $H$ be the orthocenter of $ABC$. Point
$E$ is the midpoint of $AC$ and point $D$ lies on the side $BC$ such that $3CD = BC$. Prove that
$BE \perp HD$.
[i]Proposed by Tran Quang Hung - Vietnam[/i]
Two circles intersect at points $A$ and $B$. The minor arcs $AB$ measure $30^\circ$ on one circle and $60^\circ$ on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?
$\textbf{(A) }2\qquad
\textbf{(B) }1+\sqrt3\qquad
\textbf{(C) }3\qquad
\textbf{(D) }2+\sqrt3\qquad
\textbf{(E) }4\qquad$
Right triangle $ABC$ has hypotenuse $AB = 26$, and the inscribed circle of $ABC$ has radius $5$. The largest possible value of $BC$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are both positive integers. Find $100m + n$.
[i]Proposed by Jason Xia[/i]
Triangle $ABC$ has $AB=27$, $AC=26$, and $BC=25$. Let $I$ denote the intersection of the internal angle bisectors of $\triangle ABC$. What is $BI$?
$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 5+\sqrt{26}+3\sqrt{3}\qquad\textbf{(C)}\ 3\sqrt{26}\qquad\textbf{(D)}\ \frac{2}{3}\sqrt{546}\qquad\textbf{(E)}\ 9\sqrt{3} $
Let a right cone of the base radius $r=3$ and height greater than $6$ be inscribed in a sphere of radius $R=6$. The volume of the cone can be expressed as $\pi(a\sqrt{b}+c)$, where $b$ is square free. Find $a+b+c$.
Let $ A$, $ B$, and $ C$ be three distinct points on the graph of $ y\equal{}x^2$ such that line $ AB$ is parallel to the $ x$-axis and $ \triangle{ABC}$ is a right triangle with area $ 2008$. What is the sum of the digits of the $ y$-coordinate of $ C$?
$ \textbf{(A)}\ 16 \qquad
\textbf{(B)}\ 17 \qquad
\textbf{(C)}\ 18 \qquad
\textbf{(D)}\ 19 \qquad
\textbf{(E)}\ 20$
Let $ABC$ be a scalene triangle, and let $X, Y , Z$ be points on sides $BC, CA, AB,$
respectively. Let $I$ and $O$ denote the incenter and circumcenter, respectively, of triangle
$ABC.$ Suppose that\[ \frac{BX-CX}{BA-CA}=\frac{CY-AY}{CB-AB} = \frac{AZ-BZ}{AC-BC}.\]
Prove that there exists a point $P$ on line $IO$ such that $PX \perp BC$, $PY \perp CA$, and $PZ
\perp AB.$
Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$.
[i]Proposed by El Salvador[/i]
In the complex plane, let $A$ be the set of solutions to $z^3 - 8 = 0$ and let $B$ be the set of solutions to $z^3 - 8z^2 - 8z + 64 = 0$. What is the greatest distance between a point of $A$ and a point of $B?$
$\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 2\sqrt{21} \qquad \textbf{(E) } 9 + \sqrt{3}$
Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure at right). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$.
[asy]
size(200);
pair A=origin, B=(7,0), C=(3.2,15), D=midpoint(B--C), F=(3,0), P=intersectionpoint(C--F, A--D), ex=B+40*dir(B--P), E=intersectionpoint(B--ex, A--C);
draw(A--B--C--A--D^^C--F^^B--E);
pair point=P;
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));
label("$P$", P, dir(0));[/asy]
A clueless ant makes the following route: starting at point $ A $ goes $ 1$ cm north, then $ 2$ cm east, then $ 3$ cm south, then $ 4$ cm west, immediately $ 5$ cm north, continues $ 6$ cm east, and so on, finally $ 41$ cm north and ends in point $ B $. Calculate the distance between $ A $ and $ B $ (in a straight line).
On a circle with diameter $AC$, let $B$ be an arbitrary point distinct from $A$ and $C$. Points $M, N$ are the midpoints of chords $AB, BC$, and points $P, Q$ are the midpoints of smaller arcs restricted by these chords. Lines $AQ$ and $BC$ meet at point $K$, and lines $CP$ and $AB$ meet at point $L$. Prove that lines $MQ, NP$ and $KL$ concur.