Let $ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ at the points $A_1$, $B_1$, $C_1$, respectively. Prove that $A_1B_1 \cdot B_1C_1 \cdot C_1A_1 \ge A_1B \cdot B_1C \cdot C_1A$.

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]

Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC=2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle{ABC}$ at a point $D\ne A$. Find the length of segment $CD$. [i]Ray Li.[/i]

In quadrilateral $ABCD$, $\angle DAC = 98^{\circ}$, $\angle DBC = 82^\circ$, $\angle BCD = 70^\circ$, and $BC = AD$. Find $\angle ACD.$

Let $ ABCD$ be a cyclic quadrilateral. The side lengths of $ ABCD$ are distinct integers less than $ 15$ such that $ BC\cdot CD\equal{}AB\cdot DA$. What is the largest possible value of $ BD$? $ \textbf{(A)}\ \sqrt{\frac{325}{2}} \qquad \textbf{(B)}\ \sqrt{185} \qquad \textbf{(C)}\ \sqrt{\frac{389}{2}} \qquad \textbf{(D)}\ \sqrt{\frac{425}{2}} \qquad \textbf{(E)}\ \sqrt{\frac{533}{2}}$

$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.

Sides $AB,~ BC,$ and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4,~ 5,$ and $20$, respectively. If vertex angles $B$ and $C$ are obtuse and $\sin C = - \cos B =\frac{3}{5} $, then side $AD$ has length $\textbf{(A) }24\qquad\textbf{(B) }24.5\qquad\textbf{(C) }24.6\qquad\textbf{(D) }24.8\qquad\textbf{(E) }25$ [size=70]*A polygon is called “simple” if it is not self intersecting.[/size]

Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$

In triangle $ABC$ let $A'$, $B'$, $C'$ respectively be the midpoints of the sides $BC$, $CA$, $AB$. Furthermore let $L$, $M$, $N$ be the projections of the orthocenter on the three sides $BC$, $CA$, $AB$, and let $k$ denote the nine-point circle. The lines $AA'$, $BB'$, $CC'$ intersect $k$ in the points $D$, $E$, $F$. The tangent lines on $k$ in $D$, $E$, $F$ intersect the lines $MN$, $LN$ and $LM$ in the points $P$, $Q$, $R$. Prove that $P$, $Q$ and $R$ are collinear.

Convex quadrilateral $ABCD$ has sides $AB=BC=7$, $CD=5$, and $AD=3$. Given additionally that $m\angle ABC=60^\circ$, find $BD$.

Let $A_1, A_2, \ldots, A_n$ be an $n$ -sided regular polygon. If $\frac{1}{A_1 A_2} = \frac{1}{A_1 A_3} + \frac{1}{A_1A_4}$, find $n$.

Quadrilateral $ABCD$ is both cyclic and circumscribed. Its incircle touches its sides $AB$ and $CD$ at points $X$ and $Y$, respectively. The perpendiculars to $AB$ and $CD$ drawn at $A$ and $D$, respectively, meet at point $U$; those drawn at $X$ and $Y$ meet at point $V$, and finally, those drawn at $B$ and $C$ meet at point $W$. Prove that points $U$, $V$ and $W$ are collinear. [i]Proposed by A. Golovanov[/i]

Find all ordered triples $ (x, y, z)$ of real numbers such that \[ 5 \left(x \plus{} \frac{1}{x} \right) \equal{} 12 \left(y \plus{} \frac{1}{y} \right) \equal{} 13 \left(z \plus{} \frac{1}{z} \right),\] and \[ xy \plus{} yz \plus{} zy \equal{} 1.\]

Let $ABC$ be a triangle with $\angle{BAC} = 40^{\circ}$ and $\angle{ABC}=60^{\circ}$. Let $D$ and $E$ be the points lying on the sides $AC$ and $AB$, respectively, such that $\angle{CBD} = 40^{\circ}$ and $\angle{BCE} = 70^{\circ}$. Let $F$ be the point of intersection of the lines $BD$ and $CE$. Show that the line $AF$ is perpendicular to the line $BC$.

The triangle $ABC$ has $CA = CB$. $P$ is a point on the circumcircle between $A$ and $B$ (and on the opposite side of the line $AB$ to $C$). $D$ is the foot of the perpendicular from $C$ to $PB$. Show that $PA + PB = 2 \cdot PD$.

Let $ABCD$ be a tetrahedron with $\angle BAD = 60^{\cdot}$, $\angle BAC = 40^{\cdot}$, $\angle ABD = 80^{\cdot}$, $\angle ABC = 70^{\cdot}$. Prove that the lines $AB$ and $CD$ are perpendicular.

Consider a triangle $ABC$ and $I$ its incenter. The line $(AI)$ meets the circumcircle of $ABC$ in $D$. Let $E$ and $F$ be the orthogonal projections of $I$ on $(BD)$ and $(CD)$ respectively. Assume that $IE+IF=\frac{1}{2}AD$. Calculate $\angle{BAC}$. [color=red][Moderator edited: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=5088 .][/color]

In $\triangle A B C$, let $D, E$, and $F$ be the midpoints of the sides of the triangle, and let $P, Q,$ and $R$ be the midpoints of the corresponding medians, $AD ,B E,$ and $C F$, respectively, as shown in the figure at the right. Prove that the value of \[\frac{AQ^2 + A R^2 + B P^2 + B R^2 + C P^2+ C Q^2 }{A B^2 + B C^2 + C A^2}\] does not depend on the shape of $\triangle A B C$ and find that value. [asy] defaultpen(linewidth(0.7)+fontsize(10));size(200); pair A=origin, B=(14,0), C=(9,12), D=midpoint(C--B), E=midpoint(C--A), F=midpoint(A--B), R=midpoint(C--F), P=midpoint(D--A), Q=midpoint(E--B); draw(A--B--C--A, linewidth(1)); draw(A--D^^B--E^^C--F); draw(B--R--A--Q--C--P--cycle, dashed); pair point=centroid(A,B,C); 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(40)*dir(point--P)); label("$Q$", Q, dir(40)*dir(point--Q)); label("$R$", R, dir(40)*dir(point--R)); dot(P^^Q^^R);[/asy]

Let ABCD be a convex quadrilateral such that $AB + CD = \sqrt{2}AC$ and $BC + DA = \sqrt{2}BD$. Prove that ABCD is a parallelogram.

A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form \[ a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}, \] where $a$, $b$, $c$, and $d$ are positive integers. Find $a + b + c + d$.

Determine whether or not there exist 15 integers $m_1,\ldots,m_{15}$ such that~ $$\displaystyle \sum_{k=1}^{15}\,m_k\cdot\arctan(k) = \arctan(16). \eqno(1)$$ (Proposed by Gerhard Woeginger, Eindhoven University of Technology)

Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$. Prove that $\angle {APB}=\angle {APD}$

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?

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $AC=15$. Let $M$ be the midpoint of $BC$ and let $\Gamma$ be the circle passing through $A$ and tangent to line $BC$ at $M$. Let $\Gamma$ intersect lines $AB$ and $AC$ at points $D$ and $E$, respectively, and let $N$ be the midpoint of $DE$. Suppose line $MN$ intersects lines $AB$ and $AC$ at points $P$ and $O$, respectively. If the ratio $MN:NO:OP$ can be written in the form $a:b:c$ with $a,b,c$ positive integers satisfying $\gcd(a,b,c)=1$, find $a+b+c$. [i]James Tao[/i]

Show that the number$$4\sin\frac{\pi}{34}\left(\sin\frac{3\pi}{34}+\sin\frac{7\pi}{34}+\sin\frac{11\pi}{34}+\sin\frac{15\pi}{34}\right)$$ is an integer and determine it.