Found problems: 1546
2014 Contests, 3
Let $B$ and $C$ be two fixed points on a circle centered at $O$ that are not diametrically opposed. Let $A$ be a variable point on the circle distinct from $B$ and $C$ and not belonging to the perpendicular bisector of $BC$. Let $H$ be the orthocenter of $\triangle ABC$, and $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ intersects the circle again at $D$, and finally, $NM$ and $OD$ intersect at $P$. Determine the locus of points $P$ as $A$ moves around the circle.
2011 Rioplatense Mathematical Olympiad, Level 3, 2
Let $ABC$ an acute triangle and $H$ its orthocenter. Let $E$ and $F$ be the intersection of lines $BH$ and $CH$ with $AC$ and $AB$ respectively, and let $D$ be the intersection of lines $EF$ and $BC$. Let $\Gamma_1$ be the circumcircle of $AEF$, and $\Gamma_2$ the circumcircle of $BHC$. The line $AD$ intersects $\Gamma_1$ at point $I \neq A$. Let $J$ be the feet of the internal bisector of $\angle{BHC}$ and $M$ the midpoint of the arc $\stackrel{\frown}{BC}$ from $\Gamma_2$ that contains the point $H$. The line $MJ$ intersects $\Gamma_2$ at point $N \neq M$. Show that the triangles $EIF$ and $CNB$ are similar.
1985 IMO Longlists, 93
The sphere inscribed in tetrahedron $ABCD$ touches the sides $ABD$ and $DBC$ at points $K$ and $M$, respectively. Prove that $\angle AKB = \angle DMC$.
Indonesia MO Shortlist - geometry, g4.8
Given an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Let $K$ be a point inside $ABC$ which is not $O$ nor $H$. Point $L$ and $M$ are located outside the triangle $ABC$ such that $AKCL$ and $AKBM$ are parallelogram. At last, let $BL$ and $CM$ intersects at $N$, and let $J$ be the midpoint of $HK$. Show that $KONJ$ is also a parallelogram.
[i]Raja Oktovin, Pekanbaru[/i]
Brazil L2 Finals (OBM) - geometry, 2003.5
Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.
2003 Bulgaria National Olympiad, 2
Let $H$ be an arbitrary point on the altitude $CP$ of the acute triangle $ABC$. The lines $AH$ and $BH$ intersect $BC$ and $AC$ in $M$ and $N$, respectively.
[list]
(a) Prove that $\angle NPC =\angle MPC$.
(b) Let $O$ be the common point of $MN$ and $CP$. An arbitrary line through $O$ meets the sides of quadrilateral $CNHM$ in $D$ and $E$. Prove that $\angle EPC =\angle DPC$.
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2012 China Second Round Olympiad, 1
Let $P$ be a point on the graph of the function $y=x+\frac{2}{x}(x>0)$. $PA,PB$ are perpendicular to line $y=x$ and $x=0$, respectively, the feet of perpendicular being $A$ and $B$. Find the value of $\overrightarrow{PA}\cdot \overrightarrow{PB}$.
2006 Italy TST, 1
The circles $\gamma_1$ and $\gamma_2$ intersect at the points $Q$ and $R$ and internally touch a circle $\gamma$ at $A_1$ and $A_2$ respectively. Let $P$ be an arbitrary point on $\gamma$. Segments $PA_1$ and $PA_2$ meet $\gamma_1$ and $\gamma_2$ again at $B_1$ and $B_2$ respectively.
a) Prove that the tangent to $\gamma_{1}$ at $B_{1}$ and the tangent to $\gamma_{2}$ at $B_{2}$ are parallel.
b) Prove that $B_{1}B_{2}$ is the common tangent to $\gamma_{1}$ and $\gamma_{2}$ iff $P$ lies on $QR$.
1988 China Team Selection Test, 2
Let $ABCD$ be a trapezium $AB // CD,$ $M$ and $N$ are fixed points on $AB,$ $P$ is a variable point on $CD$. $E = DN \cap AP$, $F = DN \cap MC$, $G = MC \cap PB$, $DP = \lambda \cdot CD$. Find the value of $\lambda$ for which the area of quadrilateral $PEFG$ is maximum.
2011 Tuymaada Olympiad, 4
The Duke of Squares left to his three sons a square estate, $100\times 100$ square miles, made up of ten thousand $1\times 1$ square mile square plots. The whole estate was divided among his sons as follows. Each son was assigned a point inside the estate. A $1\times 1$ square plot was bequeathed to the son whose assigned point was closest to the center of this square plot. Is it true that, irrespective of the choice of assigned points, each of the regions bequeathed to the sons is connected (that is, there is a path between every two of its points, never leaving the region)?
2014 Saudi Arabia BMO TST, 4
Let $ABC$ be a triangle with $\angle B \le \angle C$, $I$ its incenter and $D$ the intersection point of line $AI$ with side $BC$. Let $M$ and $N$ be points on sides $BA$ and $CA$, respectively, such that $BM = BD$ and $CN = CD$. The circumcircle of triangle $CMN$ intersects again line $BC$ at $P$. Prove that quadrilateral $DIMP$ is cyclic.
1970 IMO Longlists, 45
Let $M$ be an interior point of tetrahedron $V ABC$. Denote by $A_1,B_1, C_1$ the points of intersection of lines $MA,MB,MC$ with the planes $VBC,V CA,V AB$, and by $A_2,B_2, C_2$ the points of intersection of lines $V A_1, VB_1, V C_1$ with the sides $BC,CA,AB$.
[b](a)[/b] Prove that the volume of the tetrahedron $V A_2B_2C_2$ does not exceed one-fourth of the volume of $V ABC$.
[b](b)[/b] Calculate the volume of the tetrahedron $V_1A_1B_1C_1$ as a function of the volume of $V ABC$, where $V_1$ is the point of intersection of the line $VM$ with the plane $ABC$, and $M$ is the barycenter of $V ABC$.
2012 Finnish National High School Mathematics Competition, 1
A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is $5:9$. Compute the ratio of the secant line versus circle radius.
2006 Pan African, 1
Let $AB$ and $CD$ be two perpendicular diameters of a circle with centre $O$. Consider a point $M$ on the diameter $AB$, different from $A$ and $B$. The line $CM$ cuts the circle again at $N$. The tangent at $N$ to the circle and the perpendicular at $M$ to $AM$ intersect at $P$. Show that $OP = CM$.
2010 Malaysia National Olympiad, 7
A line segment of length 1 is given on the plane. Show that a line segment of length $\sqrt{2010}$ can be constructed using only a straightedge and a compass.
2011 Kyrgyzstan National Olympiad, 1
For a given chord $MN$ of a circle discussed the triangle $ABC$, whose base is the diameter $AB$ of this circle,which do not intersect the $MN$, and the sides $AC$ and $BC$ pass through the ends of $M$ and $N$ of the chord $MN$. Prove that the heights of all such triangles $ABC$ drawn from the vertex $C$ to the side $AB$, intersect at one point.
1970 Regional Competition For Advanced Students, 3
$E_1$ and $E_2$ are parallel planes and their distance is $p$.
(a) How long is the seitenkante of the regular octahedron such that a side lies in $E_1$ and another in $E_2$?
(b) $E$ is a plane between $E_1$ and $E_2$, parallel to $E_1$ and $E_2$, so that its distances from $E_1$ and $E_2$ are in ratio $1:2$
Draw the intersection figure of $E$ and the octahedron for $P=4\sqrt{\frac32}$ cm and justifies, why the that figure must look in such a way
2013 Romania Team Selection Test, 2
Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$
1995 Spain Mathematical Olympiad, 3
A line through the centroid G of the triangle ABC intersects the side AB at P and the side AC at Q Show that $\frac{PB}{PA} \cdot \frac{QC}{QA} \leq \frac{1}{4}$.
Sorry for Triple-Posting. If possible, please merge the solutions to one document.
I think there was an error because it may have automatically triple-posted.
2011 Sharygin Geometry Olympiad, 6
Two unit circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. $M$ is an arbitrary point of $\omega_1$, $N$ is an arbitrary point of $\omega_2$. Two unit circles $\omega_3$ and $\omega_4$ pass through both points $M$ and $N$. Let $C$ be the second common point of $\omega_1$ and $\omega_3$, and $D$ be the second common point of $\omega_2$ and $\omega_4$. Prove that $ACBD$ is a parallelogram.
2014 Portugal MO, 5
Let $[ABCD]$ be a convex quadrilateral with area $2014$, and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have area $1$. Let $R$ be the intersection of $[AC]$ and $[PQ]$. Determine $\frac{\overline{RC}}{\overline{RA}}$.
1998 IberoAmerican Olympiad For University Students, 2
In a plane there is an ellipse $E$ with semiaxis $a,b$. Consider all the triangles inscribed in $E$ such that at least one of its sides is parallel to one of the axis of $E$.
Find both the locus of the centroid of all such triangles and its area.
2008 Sharygin Geometry Olympiad, 24
(I.Bogdanov, 11) Let $ h$ be the least altitude of a tetrahedron, and $ d$ the least distance between its opposite edges. For what values of $ t$ the inequality $ d>th$ is possible?
1989 Vietnam National Olympiad, 3
Let be given a parallelepiped $ ABCD.A'B'C'D'$. Show that if a line $ \Delta$ intersects three of the lines $ AB'$, $ BC'$, $ CD'$, $ DA'$, then it intersects also the fourth line.
2011 Morocco National Olympiad, 4
Let $ABCD$ be a convex quadrilateral with angles $\angle ABC$ and $\angle BCD$ not less than $120^{\circ}$. Prove that
\[AC + BD> AB+BC+CD\]