Olympiad problems

1983 IMO Longlists, 73

Let $ABC$ be a nonequilateral triangle. Prove that there exist two points $P$ and $Q$ in the plane of the triangle, one in the interior and one in the exterior of the circumcircle of $ABC$, such that the orthogonal projections of any of these two points on the sides of the triangle are vertices of an equilateral triangle.

2006 Kyiv Mathematical Festival, 3

Tags: trig identities , incenter , trigonometry , geometry , law of sines , circumcircle

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.

2021-IMOC qualification, G1

Tags: incenter , projection , circumcenter , circumcircle , geometry

Let $O$ be the circumcenter and $I$ be the incenter of $\vartriangle$, $P$ is the reflection from $I$ through $O$, the foot of perpendicular from $P$ to $BC,CA,AB$ is $X,Y,Z$, respectively. Prove that $AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2$.

2017 Saudi Arabia IMO TST, 1

Tags: geometry , orthocenter , right angle , circumcircle , reflection

Let $ABC$ be a triangle inscribed in circle $(O),$ with its altitudes $BE, CF$ intersect at orthocenter $H$ ($E \in AC, F \in AB$). Let $M$ be the midpoint of $BC, K$ be the orthogonal projection of $H$ on $AM$. $EF$ intersects $BC$ at $P$. Let $Q$ be the intersection of tangent of $(O)$ which passes through $A$ with $BC, T$ be the reflection of $Q$ through $P$. Prove that $\angle OKT = 90^o$.

2005 Croatia National Olympiad, 4

Tags: trigonometry , circumcircle , geometry

The circumradius $R$ of a triangle with side lengths $a, b, c$ satisﬁes $R =\frac{a\sqrt{bc}}{b+c}$. Find the angles of the triangle.

2004 Germany Team Selection Test, 2

Tags: geometry , parallelogram , reflection , circumcircle , geometric transformation , perpendicular bisector

Let two chords $AC$ and $BD$ of a circle $k$ meet at the point $K$, and let $O$ be the center of $k$. Let $M$ and $N$ be the circumcenters of triangles $AKB$ and $CKD$. Show that the quadrilateral $OMKN$ is a parallelogram.

2016 Saudi Arabia IMO TST, 1

Tags: orthocenter , circumcircle , geometry

Let $ABC$ be a triangle inscribed in the circle $(O)$. The bisector of $\angle BAC$ cuts the circle $(O)$ again at $D$. Let $DE$ be the diameter of $(O)$. Let $G$ be a point on arc $AB$ which does not contain $C$. The lines $GD$ and $BC$ intersect at $F$. Let $H$ be a point on the line $AG$ such that $FH \parallel AE$. Prove that the circumcircle of triangle $HAB$ passes through the orthocenter of triangle $HAC$.

2007 IMO Shortlist, 2

Tags: reflection , circumcircle , geometry , mixtilinear incircle , cyclic quadrilateral

Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$. [i]Author: Farzan Barekat, Canada[/i]

1999 Austrian-Polish Competition, 8

Tags: midpoint , perpendicular , geometry , circumcircle , perpendicular bisector

Let $P,Q,R$ be points on the same side of a line $g$ in the plane. Let $M$ and $N$ be the feet of the perpendiculars from $P$ and $Q$ to $g$ respectively. Point $S$ lies between the lines $PM$ and $QN$ and satisfies and satisfies $PM = PS$ and $QN = QS$. The perpendicular bisectors of $SM$ and $SN$ meet in a point $R$. If the line $RS$ intersects the circumcircle of triangle $PQR$ again at $T$, prove that $S$ is the midpoint of $RT$.

2013 Korea National Olympiad, 6

Tags: geometry , circumcircle

Let $ O $ be circumcenter of triangle $ABC$. For a point $P$ on segmet $BC$, the circle passing through $ P, B $ and tangent to line $AB $ and the circle passing through $ P, C $ and tangent to line $AC $ meet at point $ Q ( \ne P ) $. Let $ D, E $ be foot of perpendicular from $Q$ to $ AB, AC$. ($D \ne B, E \ne C $) Two lines $DE $ and $ BC $ meet at point $R$. Prove that $ O, P, Q $ are collinear if and only if $ A, R, Q $ are collinear.

2007 Iran MO (3rd Round), 5

Tags: ratio , geometry , trapezoid , rectangle , search , parallelogram , circumcircle

Let $ ABC$ be a triangle. Squares $ AB_{c}B_{a}C$, $ CA_{b}A_{c}B$ and $ BC_{a}C_{b}A$ are outside the triangle. Square $ B_{c}B_{c}'B_{a}'B_{a}$ with center $ P$ is outside square $ AB_{c}B_{a}C$. Prove that $ BP,C_{a}B_{a}$ and $ A_{c}B_{c}$ are concurrent.

2008 Bosnia Herzegovina Team Selection Test, 2

Tags: geometry , trigonometry , radical axis , power of a point , circumcircle

Let $ AD$ be height of triangle $ \triangle ABC$ and $ R$ circumradius. Denote by $ E$ and $ F$ feet of perpendiculars from point $ D$ to sides $ AB$ and $ AC$. If $ AD\equal{}R\sqrt{2}$, prove that circumcenter of triangle $ \triangle ABC$ lies on line $ EF$.

2017 ELMO Problems, 2

Tags: orthocenter , geometry , circumcircle

Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$ [i]Proposed by Michael Ren[/i]

2006 Romania National Olympiad, 3

Tags: circumcircle , geometry

We have a quadrilateral $ABCD$ inscribed in a circle of radius $r$, for which there is a point $P$ on $CD$ such that $CB=BP=PA=AB$. (a) Prove that there are points $A,B,C,D,P$ which fulfill the above conditions. (b) Prove that $PD=r$. [i]Virgil Nicula[/i]

2015 Brazil National Olympiad, 6

Tags: circumcircle , geometry

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 $\triangle ABC$.

1998 National High School Mathematics League, 1

Tags: incenter , circumcircle , geometry

Circumcenter and incentre of $\triangle ABC$ are $O,I$. $AD$ is the height on side $BC$. If $I$ is on line $OC$, prove that the radius of circumcircle and escribed circle (in \angle BAC) are equal.

2016 German National Olympiad, 3

Tags: orthocenter , parallel , excenter , geometry , circumcircle

Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$. Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.

2017 Balkan MO Shortlist, G5

Tags: projective geometry , concurrency , geometry , circumcircle

Let $ABC$ be an acute angled triangle with orthocenter $H$. centroid $G$ and circumcircle $\omega$. Let $D$ and $M$ respectively be the intersection of lines $AH$ and $AG$ with side $BC$. Rays $MH$ and $DG$ interect $ \omega$ again at $P$ and $Q$ respectively. Prove that $PD$ and $QM$ intersect on $\omega$.

2010 Iran Team Selection Test, 8

Tags: trapezoid , circumcircle , geometry

Let $ABC$ an isosceles triangle and $BC>AB=AC$. $D,M$ are respectively midpoints of $BC, AB$. $X$ is a point such that $BX\perp AC$ and $XD||AB$. $BX$ and $AD$ meet at $H$. If $P$ is intersection point of $DX$ and circumcircle of $AHX$ (other than $X$), prove that tangent from $A$ to circumcircle of triangle $AMP$ is parallel to $BC$.

2010 Sharygin Geometry Olympiad, 2

Tags: circumcircle , geometry , vertices

Two intersecting triangles are given. Prove that at least one of their vertices lies inside the circumcircle of the other triangle. (Here, the triangle is considered the part of the plane bounded by a closed three-part broken line, a point lying on a circle is considered to be lying inside it.)

2017 Singapore MO Open, 1

Tags: midpoint , circles , incircle , circumcircle , geometry

The incircle of $\vartriangle ABC$ touches the sides $BC,CA,AB$ at $D,E,F$ respectively. A circle through $A$ and $B$ encloses $\vartriangle ABC$ and intersects the line $DE$ at points $P$ and $Q$. Prove that the midpoint of $AB$ lies on the circumircle of $\vartriangle PQF$.

2012 ELMO Shortlist, 7

Tags: circumcircle , geometry , inversion , trigonometry

Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$, let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$, and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$. Show that $\angle QPA + \angle OQB = 90^{\circ}$. [i]Alex Zhu.[/i]

2017-IMOC, G4

Tags: orthocenter , circumcircle , geometry , collinear

Given an acute $\vartriangle ABC$ with orthocenter $H$. Let $M_a$ be the midpoint of $BC. M_aH$ intersects the circumcircle of $\vartriangle ABC$ at $X_a$ and $AX_a$ intersects $BC$ at $Y_a$. Define $Y_b, Y_c$ in a similar way. Prove that $Y_a, Y_b,Y_c$ are collinear. [img]https://2.bp.blogspot.com/-yjISBHtRa0s/XnSKTrhhczI/AAAAAAAALds/e_rvs9glp60L1DastlvT0pRFyP7GnJnCwCK4BGAYYCw/s320/imoc2017%2Bg4.png[/img]

2013 National Olympiad First Round, 29

Tags: reflection , geometric transformation , circumcircle , geometry

Let $O$ be the circumcenter of triangle $ABC$ with $|AB|=5$, $|BC|=6$, $|AC|=7$. Let $A_1$, $B_1$, $C_1$ be the reflections of $O$ over the lines $BC$, $AC$, $AB$, respectively. What is the distance between $A$ and the circumcenter of triangle $A_1B_1C_1$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ \sqrt {29} \qquad\textbf{(C)}\ \dfrac {19}{2\sqrt 6} \qquad\textbf{(D)}\ \dfrac {35}{4\sqrt 6} \qquad\textbf{(E)}\ \sqrt {\dfrac {35}3} $

1991 IberoAmerican, 6

Tags: geometry , parallelogram , quadratic , circumcircle , algebra

Let $M$, $N$ and $P$ be three non-collinear points. Construct using straight edge and compass a triangle for which $M$ and $N$ are the midpoints of two of its sides, and $P$ is its orthocenter.