Found problems: 2023
2014 ELMO Shortlist, 5
Let $P$ be a point in the interior of an acute triangle $ABC$, and let $Q$ be its isogonal conjugate. Denote by $\omega_P$ and $\omega_Q$ the circumcircles of triangles $BPC$ and $BQC$, respectively. Suppose the circle with diameter $\overline{AP}$ intersects $\omega_P$ again at $M$, and line $AM$ intersects $\omega_P$ again at $X$. Similarly, suppose the circle with diameter $\overline{AQ}$ intersects $\omega_Q$ again at $N$, and line $AN$ intersects $\omega_Q$ again at $Y$.
Prove that lines $MN$ and $XY$ are parallel.
(Here, the points $P$ and $Q$ are [i]isogonal conjugates[/i] with respect to $\triangle ABC$ if the internal angle bisectors of $\angle BAC$, $\angle CBA$, and $\angle ACB$ also bisect the angles $\angle PAQ$, $\angle PBQ$, and $\angle PCQ$, respectively. For example, the orthocenter is the isogonal conjugate of the circumcenter.)
[i]Proposed by Sammy Luo[/i]
2011 ELMO Shortlist, 2
Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$.
[i]David Yang.[/i]
1996 All-Russian Olympiad, 6
In isosceles triangle $ABC$ ($AB = BC$) one draws the angle bisector $CD$. The perpendicular to $CD$ through the center of the circumcircle of $ABC$ intersects $BC$ at $E$. The parallel to $CD$ through $E$ meets $AB$ at $F$. Show that $BE$ = $FD$.
[i]M. Sonkin[/i]
2008 Iran MO (2nd Round), 2
Let $I_a$ be the $A$-excenter of $\Delta ABC$ and the $A$-excircle of $\Delta ABC$ be tangent to the lines $AB,AC$ at $B',C'$, respectively. $ I_aB,I_aC$ meet $B'C'$ at $P,Q$, respectively. $M$ is the meet point of $BQ,CP$. Prove that the length of the perpendicular from $M$ to $BC$ is equal to $r$ where $r$ is the radius of incircle of $\Delta ABC$.
2008 Canada National Olympiad, 1
$ ABCD$ is a convex quadrilateral for which $ AB$ is the longest side. Points $ M$ and $ N$ are located on sides $ AB$ and $ BC$ respectively, so that each of the segments $ AN$ and $ CM$ divides the quadrilateral into two parts of equal area. Prove that the segment $ MN$ bisects the diagonal $ BD$.
2010 ELMO Problems, 1
Determine all (not necessarily finite) sets $S$ of points in the plane such that given any four distinct points in $S$, there is a circle passing through all four or a line passing through some three.
[i]Carl Lian.[/i]
2019 Taiwan TST Round 1, 2
Given a convex pentagon $ ABCDE. $ Let $ A_1 $ be the intersection of $ BD $ with $ CE $ and define $ B_1, C_1, D_1, E_1 $ similarly, $ A_2 $ be the second intersection of $ \odot (ABD_1),\odot (AEC_1) $ and define $ B_2, C_2, D_2, E_2 $ similarly. Prove that $ AA_2, BB_2, CC_2, DD_2, EE_2 $ are concurrent.
[i]Proposed by Telv Cohl[/i]
1996 All-Russian Olympiad, 1
Points $E$ and $F$ are given on side $BC$ of convex quadrilateral $ABCD$ (with $E$ closer than $F$ to $B$). It is known that $\angle BAE = \angle CDF$ and $\angle EAF = \angle FDE$. Prove that $\angle FAC = \angle EDB$.
[i]M. Smurov[/i]
1998 Turkey MO (2nd round), 1
Let $D$ be the point on the base $BC$ of an isosceles $\vartriangle ABC$ triangle such that $\frac{\left| BD \right|}{\left| DC \right|}=\text{ }2$, and let $P$ be the point on the segment $\left[ AD \right]$ such that $\angle BAC=\angle BPD$. Prove that $\angle DPC=\frac{1}{2}\angle BAC$.
1996 Brazil National Olympiad, 2
Does there exist a set of $n > 2, n < \infty$ points in the plane such that no three are collinear and the circumcenter of any three points of the set is also in the set?
2006 Iran Team Selection Test, 5
Let $ABC$ be an acute angle triangle.
Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$.
Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$.
Prove that
\[ 2(PQ+QR+RP)\geq DE+EF+FD \]
2008 Sharygin Geometry Olympiad, 16
(A.Zaslavsky, 9--11) Given two circles. Their common external tangent is tangent to them at points $ A$ and $ B$. Points $ X$, $ Y$ on these circles are such that some circle is tangent to the given two circles at these points, and in similar way (external or internal). Determine the locus of intersections of lines $ AX$ and $ BY$.
2002 JBMO ShortLists, 10
Let $ ABC$ be a triangle with area $ S$ and points $ D,E,F$ on the sides $ BC,CA,AB$. Perpendiculars at points $ D,E,F$ to the $ BC,CA,AB$ cut circumcircle of the triangle $ ABC$ at points $ (D_1,D_2), (E_1,E_2), (F_1,F_2)$. Prove that:
$ |D_1B\cdot D_1C \minus{} D_2B\cdot D_2C| \plus{} |E_1A\cdot E_1C \minus{} E_2A\cdot E_2C| \plus{} |F_1B\cdot F_1A \minus{} F_2B\cdot F_2A| > 4S$
2012 Turkey MO (2nd round), 6
Let $B$ and $D$ be points on segments $[AE]$ and $[AF]$ respectively. Excircles of triangles $ABF$ and $ADE$ touching sides $BF$ and $DE$ is the same, and its center is $I$. $BF$ and $DE$ intersects at $C$. Let $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3, Q_4$ be the circumcenters of triangles $IAB, IBC, ICD, IDA, IAE, IEC, ICF, IFA$ respectively.
[b]a) [/b] Show that points $P_1, P_2, P_3, P_4$ concylic and points $Q_1, Q_2, Q_3, Q_4$ concylic.
[b]b) [/b] Denote centers of theese circles as $O_1$ and $O_2$. Prove that $O_1, O_2$ and $I$ are collinear.
2008 Sharygin Geometry Olympiad, 1
(B.Frenkin, 8) Does a regular polygon exist such that just half of its diagonals are parallel to its sides?
2007 Pre-Preparation Course Examination, 3
$ABC$ is an arbitrary triangle. $A',B',C'$ are midpoints of arcs $BC, AC, AB$. Sides of triangle $ABC$, intersect sides of triangle $A'B'C'$ at points $P,Q,R,S,T,F$. Prove that \[\frac{S_{PQRSTF}}{S_{ABC}}=1-\frac{ab+ac+bc}{(a+b+c)^{2}}\]
1997 Irish Math Olympiad, 2
A circle $ \Gamma$ is inscribed in a quadrilateral $ ABCD$. If $ \angle A\equal{}\angle B\equal{}120^{\circ}, \angle D\equal{}90^{\circ}$ and $ BC\equal{}1$, find, with proof, the length of $ AD$.
2013 Sharygin Geometry Olympiad, 6
Diagonals $AC$ and $BD$ of a trapezoid $ABCD$ meet at $P$. The circumcircles of triangles $ABP$ and $CDP$ intersect the line $AD$ for the second time at points $X$ and $Y$ respectively. Let $M$ be the midpoint of segment $XY$. Prove that $BM = CM$.
1988 Balkan MO, 3
Let $ABCD$ be a tetrahedron and let $d$ be the sum of squares of its edges' lengths. Prove that the tetrahedron can be included in a region bounded by two parallel planes, the distances between the planes being at most $\frac{\sqrt{d}}{2\sqrt{3}}$
2016 Croatia Team Selection Test, Problem 3
Let $P$ be a point inside a triangle $ABC$ such that
$$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$
Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.
2004 Serbia Team Selection Test, 1
Let ABCD be a square and K
be a circle with diameter AB. For an
arbitrary point P on side CD, segments AP and BP meet K
again at
points M and N, respectively, and lines DM and CN meet at point Q.
Prove that Q lies on the circle K
and that AQ : QB = DP : PC.
2012 Grigore Moisil Intercounty, 4
Let $ \Delta ABC$ be a triangle with $M$ the middle of the side $[BC]$.
On the line $BC$, to the left and to the right of the point $M,$ at the same distance from $M,$ let us consider $d_1$ and $d_2,$ which are perpendicular to the line BC.
The perpendicular line from $M$ to $AB$ intersects $d_1$ in $P,$ and the perpendicular line from $M$ to $AC$ intersects $d_2$ in $Q.$
Prove that \[AM\perp PQ.\]
[b]Author: Marin Bancoș
Regional Mathematical Contest GRIGORE MOISIL, Romania, Baia Mare, 2012, 9th grade[/b]
1997 Bulgaria National Olympiad, 2
Let $M$ be the centroid of $\Delta ABC$
Prove the inequality
$\sin \angle CAM + \sin\angle CBM \le \frac{2}{\sqrt 3}$
(a) if the circumscribed circle of $\Delta AMC$ is tangent to the line $AB$
(b) for any $\Delta ABC$
2005 Germany Team Selection Test, 3
Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.
2009 Romania Team Selection Test, 2
Prove that the circumcircle of a triangle contains exactly 3 points whose Simson lines are tangent to the triangle's Euler circle and these points are the vertices of an equilateral triangle.