Found problems: 2023
2005 Iran Team Selection Test, 2
Suppose there are $n$ distinct points on plane. There is circle with radius $r$ and center $O$ on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move $O$ to the baricenter of the point in the circle. Prove that location of $O$ is constant after some steps.
2008 Moldova Team Selection Test, 3
Let $ \Gamma(I,r)$ and $ \Gamma(O,R)$ denote the incircle and circumcircle, respectively, of a triangle $ ABC$. Consider all the triangels $ A_iB_iC_i$ which are simultaneously inscribed in $ \Gamma(O,R)$ and circumscribed to $ \Gamma(I,r)$. Prove that the centroids of these triangles are concyclic.
2016 Iranian Geometry Olympiad, 2
In acute-angled triangle $ABC$, altitude of $A$ meets $BC$ at $D$, and $M$ is midpoint of $AC$. Suppose that $X$ is a point such that $\measuredangle AXB = \measuredangle DXM =90^\circ$ (assume that $X$ and $C$ lie on opposite sides of the line $BM$). Show that $\measuredangle XMB = 2\measuredangle MBC$.Proposed by Davood Vakili
2024 Polish Junior MO Finals, 1
Can we find a convex quadrilateral $ABCD$ with an interior point $P$ satisfying
\[AB=AP, \quad BC=BP, \quad CD=CP, \quad \text{and} \quad DA=DP \quad ?\]
2014 All-Russian Olympiad, 4
Given a triangle $ABC$ with $AB>BC$, let $ \Omega $ be the circumcircle. Let $M$, $N$ lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. Let $K$ be the intersection of $MN$ and $AC$. Let $P$ be the incentre of the triangle $AMK$ and $Q$ be the $K$-excentre of the triangle $CNK$. If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$.
[i]M. Kungodjin[/i]
2007 France Team Selection Test, 3
Let $A,B,C,D$ be four distinct points on a circle such that the lines $(AC)$ and $(BD)$ intersect at $E$, the lines $(AD)$ and $(BC)$ intersect at $F$ and such that $(AB)$ and $(CD)$ are not parallel.
Prove that $C,D,E,F$ are on the same circle if, and only if, $(EF)\bot(AB)$.
1990 Brazil National Olympiad, 3
Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$.
2015 Brazil National Olympiad, 6
Let $\triangle ABC$ be a scalene triangle and $X$, $Y$ and $Z$ be points on the lines $BC$, $AC$ and $AB$, respectively, such that $\measuredangle AXB = \measuredangle BYC = \measuredangle CZA$. The circumcircles of $BXZ$ and $CXY$ intersect at $P$. Prove that $P$ is on the circumference which diameter has ends in the ortocenter $H$ and in the baricenter $G$ of $\triangle ABC$.
2013 Macedonia National Olympiad, 5
An arbitrary triangle ABC is given. There are 2 lines, p and q, that are not parallel to each other and they are not perpendicular to the sides of the triangle. The perpendicular lines through points A, B and C to line p we denote with $ p_a, p_b, p_c $ and the perpendicular lines to line q we denote with $ q_a, q_b, q_c $. Let the intersection points of the lines $ p_a, q_a, p_b, q_b, p_c $ and $ q_c $ with $ q_b, p_b, q_c, p_c, q_a $ and $ p_a $ are $ K, L, P, Q, N $ and $ M $. Prove that $ KL, MN $ and $ PQ $ intersect in one point.
2010 Contests, 3
We are given a cyclic quadrilateral $ABCD$ with a point $E$ on the diagonal $AC$ such that $AD=AE$ and $CB=CE$. Let $M$ be the center of the circumcircle $k$ of the triangle $BDE$. The circle $k$ intersects the line $AC$ in the points $E$ and $F$. Prove that the lines $FM$, $AD$ and $BC$ meet at one point.
[i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 3)[/i]
2006 Pre-Preparation Course Examination, 4
Find a 3rd degree polynomial whose roots are $r_a$, $r_b$ and $r_c$ where $r_a$ is the radius of the outer inscribed circle of $ABC$ with respect to $A$.
2006 CentroAmerican, 6
Let $ABCD$ be a convex quadrilateral. $I=AC\cap BD$, and $E$, $H$, $F$ and $G$ are points on $AB$, $BC$, $CD$ and $DA$ respectively, such that $EF \cap GH= I$. If $M=EG \cap AC$, $N=HF \cap AC$, show that \[\frac{AM}{IM}\cdot \frac{IN}{CN}=\frac{IA}{IC}.\]
1987 Iran MO (2nd round), 3
Let $L_1, L_2, L_3, L_4$ be four lines in the space such that no three of them are in the same plane. Let $L_1, L_2$ intersect in $A$, $L_2,L_3$ intersect in $B$ and $L_3, L_4$ intersect in $C.$ Find minimum and maximum number of lines in the space that intersect $L_1, L_2, L_3$ and $L_4.$ Justify your answer.
2001 Baltic Way, 10
In a triangle $ABC$, the bisector of $\angle BAC$ meets the side $BC$ at the point $D$. Knowing that $|BD|\cdot |CD|=|AD|^2$ and $\angle ADB=45^{\circ}$, determine the angles of triangle $ABC$.
2008 Junior Balkan Team Selection Tests - Romania, 1
Let $ ABCD$ be a convex quadrilateral with opposite side not parallel. The line through $ A$ parallel to $ BD$ intersect line $ CD$ in $ F$, but parallel through $ D$ to $ AC$ intersect line $ AB$ at $ E$. Denote by $ M,N,P,Q$ midpoints of the segments $ AC,BD,AF,DE$. Prove that lines $ MN,PQ$ and $ AD$ are concurrent.
2003 Iran MO (3rd Round), 16
Segment $ AB$ is fixed in plane. Find the largest $ n$, such that there are $ n$ points $ P_1,P_2,\dots,P_n$ in plane that triangles $ ABP_i$ are similar for $ 1\leq i\leq n$. Prove that all of $ P_i$'s lie on a circle.
2010 Singapore MO Open, 1
Let $CD$ be a chord of a circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$ perpendicular to $CD$ at $N$ with $AN > NB$. A circle $\Gamma_2$ centered at $C$ with radius $CN$ intersects $\Gamma_1$ at points $P$ and $Q$. The line $PQ$ intersects $CD$ at $M$ and $AC$ at $K$; and the extension of $NK$ meets $\Gamma_2$ at $L$. Prove that $PQ$ is perpendicular to $AL$
2002 ITAMO, 2
The plan of a house has the shape of a capital $L$, obtained by suitably placing side-by-side four squares whose sides are $10$ metres long. The external walls of the house are $10$ metres high. The roof of the house has six faces, starting at the top of the six external walls, and each face forms an angle of $30^\circ$ with respect to a horizontal plane.
Determine the volume of the house (that is, of the solid delimited by the six external walls, the six faces of the roof, and the base of the house).
2011 APMO, 3
Let $ABC$ be an acute triangle with $\angle BAC=30^{\circ}$. The internal and external angle bisectors of $\angle ABC$ meet the line $AC$ at $B_1$ and $B_2$, respectively, and the internal and external angle bisectors of $\angle ACB$ meet the line $AB$ at $C_1$ and $C_2$, respectively. Suppose that the circles with diameters $B_1B_2$ and $C_1C_2$ meet inside the triangle $ABC$ at point $P$. Prove that $\angle BPC=90^{\circ}$ .
2013 Canadian Mathematical Olympiad Qualification Repechage, 8
Let $\triangle ABC$ be an acute-angled triangle with orthocentre $H$ and circumcentre $O$. Let $R$ be the radius of the circumcircle. \begin{align*}
\text{Let }\mathit{A'}\text{ be the point on }\mathit{AO}\text{ (extended if necessary) for which }\mathit{HA'}\perp\mathit{AO}. \\
\text{Let }\mathit{B'}\text{ be the point on }\mathit{BO}\text{ (extended if necessary) for which }\mathit{HB'}\perp\mathit{BO}. \\
\text{Let }\mathit{C'}\text{ be the point on }\mathit{CO}\text{ (extended if necessary) for which }\mathit{HC'}\perp\mathit{CO}.\end{align*} Prove that $HA'+HB'+HC'<2R$
(Note: The orthocentre of a triangle is the intersection of the three altitudes of the triangle. The circumcircle of a triangle is the circle passing through the triangle’s three vertices. The circummcentre is the centre of the circumcircle.)
2002 JBMO ShortLists, 9
In triangle $ ABC,H,I,O$ are orthocenter, incenter and circumcenter, respectively. $ CI$ cuts circumcircle at $ L$. If $ AB\equal{}IL$ and $ AH\equal{}OH$, find angles of triangle $ ABC$.
2009 Turkey MO (2nd round), 2
Let $\Gamma$ be the circumcircle of a triangle $ABC,$ and let $D$ and $E$ be two points different from the vertices on the sides $AB$ and $AC,$ respectively. Let $A'$ be the second point where $\Gamma$ intersects the bisector of the angle $BAC,$ and let $P$ and $Q$ be the second points where $\Gamma$ intersects the lines $A'D$ and $A'E,$ respectively. Let $R$ and $S$ be the second points of intersection of the lines $AA'$ and the circumcircles of the triangles $APD$ and $AQE,$ respectively.
Show that the lines $DS, \: ER$ and the tangent line to $\Gamma$ through $A$ are concurrent.
2008 China Second Round Olympiad, 1
Given a convex quadrilateral with $\angle B+\angle D<180$.Let $P$ be an arbitrary point on the plane,define
$f(P)=PA*BC+PD*CA+PC*AB$.
(1)Prove that $P,A,B,C$ are concyclic when $f(P)$ attains its minimum.
(2)Suppose that $E$ is a point on the minor arc $AB$ of the circumcircle $O$ of $ABC$,such that$AE=\frac{\sqrt 3}{2}AB,BC=(\sqrt 3-1)EC,\angle ECA=2\angle ECB$.Knowing that $DA,DC$ are tangent to circle $O$,$AC=\sqrt 2$,find the minimum of $f(P)$.
2004 Germany Team Selection Test, 1
Let $ABC$ be an acute triangle, and let $M$ and $N$ be two points on the line $AC$ such that the vectors $MN$ and $AC$ are identical. Let $X$ be the orthogonal projection of $M$ on $BC$, and let $Y$ be the orthogonal projection of $N$ on $AB$. Finally, let $H$ be the orthocenter of triangle $ABC$.
Show that the points $B$, $X$, $H$, $Y$ lie on one circle.
2010 Sharygin Geometry Olympiad, 5
A point $E$ lies on the altitude $BD$ of triangle $ABC$, and $\angle AEC=90^\circ.$ Points $O_1$ and $O_2$ are the circumcenters of triangles $AEB$ and $CEB$; points $F, L$ are the midpoints of the segments $AC$ and $O_1O_2.$ Prove that the points $L,E,F$ are collinear.