Found problems: 2023
2010 APMO, 1
Let $ABC$ be a triangle with $\angle BAC \neq 90^{\circ}.$ Let $O$ be the circumcenter of the triangle $ABC$ and $\Gamma$ be the circumcircle of the triangle $BOC.$ Suppose that $\Gamma$ intersects the line segment $AB$ at $P$ different from $B$, and the line segment $AC$ at $Q$ different from $C.$ Let $ON$ be the diameter of the circle $\Gamma.$ Prove that the quadrilateral $APNQ$ is a parallelogram.
2015 Spain Mathematical Olympiad, 3
Let $ABC$ be a triangle. $M$, and $N$ points on $BC$, such that $BM=CN$, with $M$ in the interior of $BN$. Let $P$ and $Q$ be points in $AN$ and $AM$ respectively such that $\angle PMC= \angle MAB$, and $\angle QNB= \angle NAC$. Prove that $ \angle QBC= \angle PCB$.
2005 Croatia National Olympiad, 2
The incircle of a triangle $ABC$ touches $AC, BC$ , and $AB$ at $M , N$, and $R$, respectively. Let $S$ be a point on the smaller arc $MN$ and $t$ be the tangent to this arc at $S$ . The line $t$ meets $NC$ at $P$ and $MC$ at $Q$. Prove that the lines $AP, BQ, SR, MN$ have a common point.
2008 Irish Math Olympiad, 2
Circles $ S$ and $ T$ intersect at $ P$ and $ Q$, with $ S$ passing through the centre of $ T$. Distinct points $ A$ and $ B$ lie on $ S$, inside $ T$, and are equidistant from the centre of $ T$. The line $ PA$ meets $ T$ again at $ D$. Prove that $ |AD| \equal{} |PB|$.
2005 Brazil National Olympiad, 5
Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid.
Prove that these three Euler lines pass through one common point.
[i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$.
[i]Floor van Lamoen[/i]
2009 Iran MO (2nd Round), 3
Let $ ABC $ be a triangle and the point $ D $ is on the segment $ BC $ such that $ AD $ is the interior bisector of $ \angle A $. We stretch $ AD $ such that it meets the circumcircle of $ \Delta ABC $ at $ M $. We draw a line from $ D $ such that it meets the lines $ MB,MC $ at $ P,Q $, respectively ($ M $ is not between $ B,P $ and also is not between $ C,Q $).
Prove that $ \angle PAQ\geq\angle BAC $.
2007 Pre-Preparation Course Examination, 2
Let $C_{1}$, $C_{2}$ and $C_{3}$ be three circles that does not intersect and non of them is inside another. Suppose $(L_{1},L_{2})$, $(L_{3},L_{4})$ and $(L_{5},L_{6})$ be internal common tangents of $(C_{1}, C_{2})$, $(C_{1}, C_{3})$, $(C_{2}, C_{3})$. Let $L_{1},L_{2},L_{3},L_{4},L_{5},L_{6}$ be sides of polygon $AC'BA'CB'$. Prove that $AA',BB',CC'$ are concurrent.
2013 Federal Competition For Advanced Students, Part 2, 3
A square and an equilateral triangle are inscribed in a same circle. The seven vertices form a convex heptagon $S$ inscribed in the circle ($S$ might be a hexagon if two vertices coincide). For which positions of the triangle relative to the square does $S$ have the largest and smallest area, respectively?
2008 Pan African, 2
Let $C_1$ be a circle with centre $O$, and let $AB$ be a chord of the circle that is not a diameter. $M$ is the midpoint of $AB$. Consider a point $T$ on the circle $C_2$ with diameter $OM$. The tangent to $C_2$ at the point $T$ intersects $C_1$ at two points. Let $P$ be one of these points. Show that $PA^2+PB^2=4PT^2$.
2012 Paraguay Mathematical Olympiad, 3
Let $ABC$ be a triangle (right in $B$) inscribed in a semi-circumference of diameter $AC=10$. Determine the distance of the vertice $B$ to the side $AC$ if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle.
2013 Greece Team Selection Test, 2
Let $ABC$ be a non-isosceles,aqute triangle with $AB<AC$ inscribed in circle $c(O,R)$.The circle $c_{1}(B,AB)$ crosses $AC$ at $K$ and $c$ at $E$.
$KE$ crosses $c$ at $F$ and $BO$ crosses $KE$ at $L$ and $AC$ at $M$ while $AE$ crosses $BF$ at $D$.Prove that:
i)$D,L,M,F$ are concyclic.
ii)$B,D,K,M,E$ are concyclic.
2008 China Team Selection Test, 1
Let $ ABC$ be a triangle, line $ l$ cuts its sides $ BC,CA,AB$ at $ D,E,F$, respectively. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ AEF,BFD,CDE$, respectively. Prove that the orthocenter of triangle $ O_{1}O_{2}O_{3}$ lies on line $ l$.
2006 Canada National Olympiad, 5
The vertices of a right triangle $ABC$ inscribed in a circle divide the circumference into three arcs. The right angle is at $A$, so that the opposite arc $BC$ is a semicircle while arc $BC$ and arc $AC$ are supplementary. To each of three arcs, we draw a tangent such that its point of tangency is the mid point of that portion of the tangent intercepted by the extended lines $AB,AC$. More precisely, the point $D$ on arc $BC$ is the midpoint of the segment joining the points $D'$ and $D''$ where tangent at $D$ intersects the extended lines $AB,AC$. Similarly for $E$ on arc $AC$ and $F$ on arc $AB$. Prove that triangle $DEF$ is equilateral.
2012 Kyoto University Entry Examination, 2
Given a regular tetrahedron $OABC$. Take points $P,\ Q,\ R$ on the sides $OA,\ OB,\ OC$ respectively. Note that $P,\ Q,\ R$ are different from the vertices of the tetrahedron $OABC$. If $\triangle{PQR}$ is an equilateral triangle, then prove that three sides $PQ,\ QR,\ RP$ are pararell to three sides $AB,\ BC,\ CA$ respectively.
30 points
2014 Taiwan TST Round 2, 6
Let $P$ be a point inside triangle $ABC$, and suppose lines $AP$, $BP$, $CP$ meet the circumcircle again at $T$, $S$, $R$ (here $T \neq A$, $S \neq B$, $R \neq C$). Let $U$ be any point in the interior of $PT$. A line through $U$ parallel to $AB$ meets $CR$ at $W$, and the line through $U$ parallel to $AC$ meets $BS$ again at $V$. Finally, the line through $B$ parallel to $CP$ and the line through $C$ parallel to $BP$ intersect at point $Q$. Given that $RS$ and $VW$ are parallel, prove that $\angle CAP = \angle BAQ$.
2020 Bundeswettbewerb Mathematik, 3
Let $AB$ be the diameter of a circle $k$ and let $E$ be a point in the interior of $k$. The line $AE$ intersects $k$ a second time in $C \ne A$ and the line $BE$ intersects $k$ a second time in $D \ne B$.
Show that the value of $AC \cdot AE+BD\cdot BE$ is independent of the choice of $E$.
2012 Kyoto University Entry Examination, 5
Give the answer about the following propositions $(p),\ (q)$ whether they are true or not. If the answer is true, then give the proof and if the answer is false, then give the proof by giving the counter example.
$(p)$ If we can form a triangle such that one of inner angles of the triangle is $60^\circ$ by choosing 3 points from the vertices of a regular $n$-polygon, then $n$ is a multiple of 3.
$(q)$ In $\triangle{ABC},\ \triangle{ABD}$, if $AC<AD$ and $BC<BD$, then $\angle{C}>\angle{D}$.
35 points
2004 Bulgaria Team Selection Test, 1
The points $P$ and $Q$ lie on the diagonals $AC$ and $BD$, respectively, of a quadrilateral $ABCD$ such that $\frac{AP}{AC} + \frac{BQ}{BD} =1$. The line $PQ$ meets the sides $AD$ and $BC$ at points $M$ and $N$. Prove that the circumcircles of the triangles $AMP$, $BNQ$, $DMQ$, and $CNP$ are concurrent.
2009 Portugal MO, 2
Points $N$ and $M$ are on the sides $CD$ and $BC$ of square $ABCD$, respectively. The perimeter of triangle $MCN$ is equal to the double of the length of the square's side. Find $\angle MAN$.
2000 Federal Competition For Advanced Students, Part 2, 2
A trapezoid $ABCD$ with $AB \parallel CD$ is inscribed in a circle $k$. Points $P$ and $Q$ are chose on the arc $ADCB$ in the order $A-P -Q-B$. Lines $CP$ and $AQ$ meet at $X$, and lines $BP$ and $DQ$ meet at $Y$. Show that points $P,Q,X, Y$ lie on a circle.
Brazil L2 Finals (OBM) - geometry, 2001.6
An altitude of a convex quadrilateral is a line through the midpoint of a side perpendicular to the opposite side. Show that the four altitudes are concurrent iff the quadrilateral is cyclic.
2002 Tournament Of Towns, 5
An angle and a point $A$ inside it is given. Is it possible to draw through $A$ three straight lines so that on either side of the angle one of three points of intersection of these lines be the midpoint of two other points of intersection with that side?
2004 Mediterranean Mathematics Olympiad, 2
In a triangle $ABC$, the altitude from $A$ meets the circumcircle again at $T$ . Let $O$ be the circumcenter. The lines $OA$ and $OT$ intersect the side $BC$ at $Q$ and $M$, respectively. Prove that
\[\frac{S_{AQC}}{S_{CMT}} = \biggl( \frac{ \sin B}{\cos C} \biggr)^2 .\]
1995 Baltic Way, 17
Prove that there exists a number $\alpha$ such that for any triangle $ABC$ the inequality
\[ \max(h_A,h_B,h_C)\le \alpha\cdot\min(m_A,m_B,m_C)\]
where $h_A,h_B,h_C$ denote the lengths of the altitudes and $m_A,m_B,m_C$ denote the lengths of the medians. Find the smallest possible value of $\alpha$.
2012 Indonesia TST, 2
Let $\omega$ be a circle with center $O$, and let $l$ be a line not intersecting $\omega$. $E$ is a point on $l$ such that $OE$ is perpendicular with $l$. Let $M$ be an arbitrary point on $M$ different from $E$. Let $A$ and $B$ be distinct points on the circle $\omega$ such that $MA$ and $MB$ are tangents to $\omega$. Let $C$ and $D$ be the foot of perpendiculars from $E$ to $MA$ and $MB$ respectively. Let $F$ be the intersection of $CD$ and $OE$. As $M$ moves, determine the locus of $F$.