Found problems: 1581
2011 USAMO, 3
In hexagon $ABCDEF$, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy $\angle A=3\angle D$, $\angle C=3\angle F$, and $\angle E=3\angle B$. Furthermore $AB=DE$, $BC=EF$, and $CD=FA$. Prove that diagonals $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent.
2014 Contests, 3
Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.
2014 USA Team Selection Test, 1
Let $ABC$ be an acute triangle, and let $X$ be a variable interior point on the minor arc $BC$ of its circumcircle. Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to lines $CA$ and $CB$, respectively. Let $R$ be the intersection of line $PQ$ and the perpendicular from $B$ to $AC$. Let $\ell$ be the line through $P$ parallel to $XR$. Prove that as $X$ varies along minor arc $BC$, the line $\ell$ always passes through a fixed point. (Specifically: prove that there is a point $F$, determined by triangle $ABC$, such that no matter where $X$ is on arc $BC$, line $\ell$ passes through $F$.)
[i]Robert Simson et al.[/i]
2015 Kazakhstan National Olympiad, 3
A rectangle is said to be $ inscribed$ in a triangle if all its vertices lie on the sides of the triangle. Prove that the locus of the centers (the meeting points of the diagonals) of all inscribed in an acute-angled triangle rectangles are three concurrent unclosed segments.
2006 China Team Selection Test, 2
Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$.
Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.
2010 Korea National Olympiad, 2
Let $ ABCD$ be a cyclic convex quadrilateral. Let $ E $ be the intersection of lines $ AB, CD $. $ P $ is the intersection of line passing $ B $ and perpendicular to $ AC $, and line passing $ C $ and perpendicular to $ BD$. $ Q $ is the intersection of line passing $ D $ and perpendicular to $ AC $, and line passing $ A $ and perpendicular to $ BD $. Prove that three points $ E, P, Q $ are collinear.
1981 AMC 12/AHSME, 20
A ray of light originates from point $A$ and and travels in a plane, being reflected $n$ times between lines $AD$ and $CD$, before striking a point $B$ (which may be on $AD$ or $CD$) perpendicularly and retracing its path to $A$. (At each point of reflection the light makes two equal angles as indicated in the adjoining figure. The figure shows the light path for $n = 3.$) If $\measuredangle CDA = 8^\circ$, what is the largest value $n$ can have?
$\text{(A)} \ 6 \qquad \text{(B)} \ 10 \qquad \text{(C)} \ 38 \qquad \text{(D)} \ 98 \qquad \text{(E)} \ \text{There is no largest value.}$
2014 NIMO Problems, 7
Find the sum of all integers $n$ with $2 \le n \le 999$ and the following property: if $x$ and $y$ are randomly selected without replacement from the set $\left\{ 1,2,\dots,n \right\}$, then $x+y$ is even with probability $p$, where $p$ is the square of a rational number.
[i]Proposed by Ivan Koswara[/i]
2014 Indonesia MO Shortlist, G3
Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.
2013 China Team Selection Test, 2
Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.
2008 IMAC Arhimede, 5
The diagonals of the cyclic quadrilateral $ ABCD$ are intersecting at the point $ E$.
$ K$ and $ M$ are the midpoints of $ AB$ and $ CD$, respectively. Let the points $ L$ on $ BC$ and $ N$ on $ AD$ s.t.
$ EL\perp BC$ and $ EN\perp AD$.Prove that $ KM\perp LN$.
2013 Romania Team Selection Test, 2
The vertices of two acute-angled triangles lie on the same circle. The Euler circle (nine-point circle) of one of the triangles passes through the midpoints of two sides of the other triangle. Prove that the triangles have the same Euler circle.
EDIT by pohoatza (in concordance with Luis' PS): [hide=Alternate/initial version ]Let $ABC$ be a triangle with circumcenter $\Gamma$ and nine-point center $\gamma$. Let $X$ be a point on $\Gamma$ and let $Y$, $Z$ be on $\Gamma$ so that the midpoints of segments $XY$ and $XZ$ are on $\gamma$. Prove that the midpoint of $YZ$ is on $\gamma$.[/hide]
2011 All-Russian Olympiad, 2
Given is an acute angled triangle $ABC$. A circle going through $B$ and the triangle's circumcenter, $O$, intersects $BC$ and $BA$ at points $P$ and $Q$ respectively. Prove that the intersection of the heights of the triangle $POQ$ lies on line $AC$.
2007 Iran Team Selection Test, 3
Let $\omega$ be incircle of $ABC$. $P$ and $Q$ are on $AB$ and $AC$, such that $PQ$ is parallel to $BC$ and is tangent to $\omega$. $AB,AC$ touch $\omega$ at $F,E$. Prove that if $M$ is midpoint of $PQ$, and $T$ is intersection point of $EF$ and $BC$, then $TM$ is tangent to $\omega$.
[i]By Ali Khezeli[/i]
2013 Online Math Open Problems, 46
Let $ABC$ be a triangle with $\angle B - \angle C = 30^{\circ}$. Let $D$ be the point where the $A$-excircle touches line $BC$, $O$ the circumcenter of triangle $ABC$, and $X,Y$ the intersections of the altitude from $A$ with the incircle with $X$ in between $A$ and $Y$. Suppose points $A$, $O$ and $D$ are collinear. If the ratio $\frac{AO}{AX}$ can be expressed in the form $\frac{a+b\sqrt{c}}{d}$ for positive integers $a,b,c,d$ with $\gcd(a,b,d)=1$ and $c$ not divisible by the square of any prime, find $a+b+c+d$.
[i]James Tao[/i]
2012 Online Math Open Problems, 31
Let $ABC$ be a triangle inscribed in circle $\Gamma$, centered at $O$ with radius $333.$ Let $M$ be the midpoint of $AB$, $N$ be the midpoint of $AC$, and $D$ be the point where line $AO$ intersects $BC$. Given that lines $MN$ and $BO$ concur on $\Gamma$ and that $BC = 665$, find the length of segment $AD$.
[i]Author: Alex Zhu[/i]
2014 Online Math Open Problems, 17
Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$
[i]Proposed by Ray Li[/i]
2011 CentroAmerican, 2
In a scalene triangle $ABC$, $D$ is the foot of the altitude through $A$, $E$ is the intersection of $AC$ with the bisector of $\angle ABC$ and $F$ is a point on $AB$. Let $O$ the circumcenter of $ABC$ and $X=AD\cap BE$, $Y=BE\cap CF$, $Z=CF \cap AD$. If $XYZ$ is an equilateral triangle, prove that one of the triangles $OXY$, $OYZ$, $OZX$ must be equilateral.
2012 Romania Team Selection Test, 2
Let $ABCD$ be a convex circumscribed quadrilateral such that $\angle ABC+\angle ADC<180^{\circ}$ and $\angle ABD+\angle ACB=\angle ACD+\angle ADB$. Prove that one of the diagonals of quadrilateral $ABCD$ passes through the other diagonals midpoint.
2007 Balkan MO, 4
For a given positive integer $n >2$, let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that
$C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}$ are finite. Find the maximum number of points of the sets $C_{1}\cap C_{2}\cap C_{3}$.
1993 AIME Problems, 7
Three numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3$, with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
2010 Today's Calculation Of Integral, 662
In $xyz$ space, let $A$ be the solid generated by a rotation of the figure, enclosed by the curve $y=2-2x^2$ and the $x$-axis about the $y$-axis.
(1) When the solid is cut by the plane $x=a\ (|a|\leq 1)$, find the inequality which expresses the figure of the cross-section.
(2) Denote by $L$ the distance between the point $(a,\ 0,\ 0)$ and the point on the perimeter of the cross-section found in (1), find the maximum value of $L$.
(3) Find the volume of the solid by a rotation of the solid $A$ about the $x$-axis.
[i]1987 Sophia University entrance exam/Science and Technology[/i]
2008 AIME Problems, 14
Let $ \overline{AB}$ be a diameter of circle $ \omega$. Extend $ \overline{AB}$ through $ A$ to $ C$. Point $ T$ lies on $ \omega$ so that line $ CT$ is tangent to $ \omega$. Point $ P$ is the foot of the perpendicular from $ A$ to line $ CT$. Suppose $ AB \equal{} 18$, and let $ m$ denote the maximum possible length of segment $ BP$. Find $ m^{2}$.
2008 AMC 10, 19
Rectangle $ PQRS$ lies in a plane with $ PQ = RS = 2$ and $ QR = SP = 6$. The rectangle is rotated $ 90^\circ$ clockwise about $ R$, then rotated $ 90^\circ$ clockwise about the point that $ S$ moved to after the first rotation. What is the length of the path traveled by point $ P$?
${ \textbf{(A)}\ (2\sqrt3 + \sqrt5})\pi \qquad \textbf{(B)}\ 6\pi \qquad \textbf{(C)}\ (3 + \sqrt {10})\pi \qquad \textbf{(D)}\ (\sqrt3 + 2\sqrt5)\pi \\ \textbf{(E)}\ 2\sqrt {10}\pi$
2003 Rioplatense Mathematical Olympiad, Level 3, 2
Triangle $ABC$ is inscribed in the circle $\Gamma$. Let $\Gamma_a$ denote the circle internally tangent to $\Gamma$ and also tangent to sides $AB$ and $AC$. Let $A'$ denote the point of tangency of $\Gamma$ and $\Gamma_a$. Define $B'$ and $C'$ similarly. Prove that $AA'$, $BB'$ and $CC'$ are concurrent.