Olympiad problems

2008 Tuymaada Olympiad, 4

Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$. [i]Author: L. Emelyanov[/i]

2015 Cuba MO, 8

Tags: circumcircle , geometry

Let $ABC$ be an acute triangle and $D$ be the foot of the altiutude from $A$ on $BC$, $E$ and $F$ are the midpoints of $BD$ and $DC$ respectively. $O$ and $Q$ are the circumcenters of the triangles $\vartriangle BF$ and $\vartriangle ACE$ respectively. $P$ is the intersection point of $OE$ and $QF$, show that $PB = PC$.

2010 Sharygin Geometry Olympiad, 1

Tags: circumcircle , angle bisector , tangent , geometry

For a nonisosceles triangle $ABC$, consider the altitude from vertex $A$ and two bisectrices from remaining vertices. Prove that the circumcircle of the triangle formed by these three lines touches the bisectrix from vertex $A$.

1989 IMO Longlists, 14

Tags: geometry , geometric inequality , area of a triangle , circumcircle

For a triangle $ ABC,$ let $ k$ be its circumcircle with radius $ r.$ The bisectors of the inner angles $ A, B,$ and $ C$ of the triangle intersect respectively the circle $ k$ again at points $ A', B',$ and $ C'.$ Prove the inequality \[ 16Q^3 \geq 27 r^4 P,\] where $ Q$ and $ P$ are the areas of the triangles $ A'B'C'$ and $ABC$ respectively.

2011 Uzbekistan National Olympiad, 2

Tags: circumcircle , parameterization , trigonometry , geometry

Let triangle ABC with $ AB=c$ $AC=b$ $BC=a$ $R$ circumradius, $p$ half peremetr of $ABC$. I f $\frac{acosA+bcosB+ccosC}{asinA+bsinB+csinC}=\frac{p}{9R}$ then find all value of $cosA$.

2004 Germany Team Selection Test, 3

Tags: geometry , analytic geometry , circumcircle , incenter , trigonometry , ratio , conic

Given six real numbers $a$, $b$, $c$, $x$, $y$, $z$ such that $0 < b-c < a < b+c$ and $ax + by + cz = 0$. What is the sign of the sum $ayz + bzx + cxy$ ?

2017 Iran MO (3rd round), 1

Tags: geometry , circumcircle , incenter

Let $ABC$ be a triangle. Suppose that $X,Y$ are points in the plane such that $BX,CY$ are tangent to the circumcircle of $ABC$, $AB=BX,AC=CY$ and $X,Y,A$ are in the same side of $BC$. If $I$ be the incenter of $ABC$ prove that $\angle BAC+\angle XIY=180$.

2023 Romanian Master of Mathematics Shortlist, G1

Tags: trigonometry , angle chasing , concurrency , geometry , circumcircle

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. The incircle of the triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$, respectively. The circumcircle of triangle $ADI$ crosses $\omega$ again at $P$, and the lines $PE$ and $PF$ cross $\omega$ again at $X$and $Y$, respectively. Prove that the lines $AI$, $BX$ and $CY$ are concurrent.

2007 IberoAmerican, 2

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

Let $ ABC$ be a triangle with incenter $ I$ and let $ \Gamma$ be a circle centered at $ I$, whose radius is greater than the inradius and does not pass through any vertex. Let $ X_{1}$ be the intersection point of $ \Gamma$ and line $ AB$, closer to $ B$; $ X_{2}$, $ X_{3}$ the points of intersection of $ \Gamma$ and line $ BC$, with $ X_{2}$ closer to $ B$; and let $ X_{4}$ be the point of intersection of $ \Gamma$ with line $ CA$ closer to $ C$. Let $ K$ be the intersection point of lines $ X_{1}X_{2}$ and $ X_{3}X_{4}$. Prove that $ AK$ bisects segment $ X_{2}X_{3}$.

2008 Federal Competition For Advanced Students, Part 2, 3

Tags: circumcircle , geometric transformation , reflection , geometry

We are given a square $ ABCD$. Let $ P$ be a point not equal to a corner of the square or to its center $ M$. For any such $ P$, we let $ E$ denote the common point of the lines $ PD$ and $ AC$, if such a point exists. Furthermore, we let $ F$ denote the common point of the lines $ PC$ and $ BD$, if such a point exists. All such points $ P$, for which $ E$ and $ F$ exist are called acceptable points. Determine the set of all acceptable points, for which the line $ EF$ is parallel to $ AD$.

Novosibirsk Oral Geo Oly IX, 2022.4

Tags: geometry , incenter , circumcircle

A point $D$ is marked on the side $AC$ of triangle $ABC$. The circumscribed circle of triangle $ABD$ passes through the center of the inscribed circle of triangle $BCD$. Find $\angle ACB$ if $\angle ABC = 40^o$.

Estonia Open Junior - geometry, 1999.2.3

Tags: circumcircle , geometry , circles , tangent circle

On the plane there are two non-intersecting circles with equal radii and with centres $O_1$ and $O_2$, line $s$ going through these centres, and their common tangent $t$. The third circle is tangent to these two circles in points $K$ and $L$ respectively, line $s$ in point $M$ and line $t$ in point $P$. The point of tangency of line $t$ and the first circle is $N$. a) Find the length of the segment $O_1O_2$. b) Prove that the points $M, K$ and $N$ lie on the same line

2017 Hong Kong TST, 1

Tags: geometry , angle bisector , circumcircle

In $\triangle ABC$, let $AD$ be the angle bisector of $\angle BAC$, with $D$ on $BC$. The perpendicular from $B$ to $AD$ intersects the circumcircle of $\triangle ABD$ at $B$ and $E$. Prove that $E$, $A$ and the circumcenter $O$ of $\triangle ABC$ are collinear.

2011 China Girls Math Olympiad, 2

Tags: geometry , circumcircle , trapezoid , rectangle , projective geometry , perpendicular bisector , geometric transformation

The diagonals $AC,BD$ of the quadrilateral $ABCD$ intersect at $E$. Let $M,N$ be the midpoints of $AB,CD$ respectively. Let the perpendicular bisectors of the segments $AB,CD$ meet at $F$. Suppose that $EF$ meets $BC,AD$ at $P,Q$ respectively. If $MF\cdot CD=NF\cdot AB$ and $DQ\cdot BP=AQ\cdot CP$, prove that $PQ\perp BC$.

2009 Saint Petersburg Mathematical Olympiad, 5

Tags: geometry , circumcircle

$O$ -circumcenter of $ABCD$. $AC$ and $BD$ intersect in $E$, $AD$ and $BC$ in $F$. $X,Y$ - midpoints of $AD$ and $BC$. $O_1$ -circumcenter of $EXY$. Prove that $OF \parallel O_1E$

Indonesia MO Shortlist - geometry, g2.6

Tags: circumcircle , geometry

Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.

2013 HMIC, 3

Tags: circumcircle , geometry

Triangle $ABC$ is inscribed in a circle $\omega$ such that $\angle A = 60^o$ and $\angle B = 75^o$. Let the bisector of angle $A$ meet $BC$ and $\omega$ at $E$ and $D$, respectively. Let the reflections of $A$ across $D$ and $C$ be $D'$ and $C'$ , respectively. If the tangent to $\omega$ at $A$ meets line $BC$ at $P$, and the circumcircle of $APD'$ meets line $AC$ at $F \ne A$, prove that the circumcircle of $C'FE$ is tangent to $BC$ at $E$.

2016 Hong Kong TST, 1

Tags: geometry , circumcircle

Let $O$ be the circumcenter of a triangle $ABC$, and let $l$ be the line going through the midpoint of the side $BC$ and is perpendicular to the bisector of $\angle BAC$. Determine the value of $\angle BAC$ if the line $l$ goes through the midpoint of the line segment $AO$.

1991 All Soviet Union Mathematical Olympiad, 555

Tags: square , circumcircle , geometry

$ABCD$ is a square. The points $X$ on the side $AB$ and $Y$ on the side $AD$ are such that $AX\cdot AY = 2 BX\cdot DY$. The lines $CX$ and $CY$ meet the diagonal $BD$ in two points. Show that these points lie on the circumcircle of $AXY$.

2007 Irish Math Olympiad, 3

Tags: circumcircle , geometric transformation , reflection , geometry

The point $ P$ is a fixed point on a circle and $ Q$ is a fixed point on a line. The point $ R$ is a variable point on the circle such that $ P,Q,$ and $ R$ are not collinear. The circle through $ P,Q,$ and $ R$ meets the line again at $ V$. Show that the line $ VR$ passes through a fixed point.

2021 IMO Shortlist, G8

Tags: circumcircle , excircle , ddit , geometry

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.

2023 Iran Team Selection Test, 2

Tags: cyclic quadrilateral , circumcircle , geometry

$ABCD$ is cyclic quadrilateral and $O$ is the center of its circumcircle. Suppose that $AD \cap BC = E$ and $AC \cap BD = F$. Circle $\omega$ is tanget to line $AC$ and $BD$. $PQ$ is a diameter of $\omega$ that $F$ is orthocenter of $EPQ$. Prove that line $OE$ is passing through center of $\omega$ [i]Proposed by Mahdi Etesami Fard [/i]

2010 Postal Coaching, 5

Tags: rectangle , geometric transformation , reflection , circumcircle , angle bisector , geometry

A point $P$ lies on the internal angle bisector of $\angle BAC$ of a triangle $\triangle ABC$. Point $D$ is the midpoint of $BC$ and $PD$ meets the external angle bisector of $\angle BAC$ at point $E$. If $F$ is the point such that $PAEF$ is a rectangle then prove that $PF$ bisects $\angle BFC$ internally or externally.

2009 Balkan MO Shortlist, G6

Tags: geometry , geometric transformation , reflection , circumcircle , rotation , trapezoid , trigonometry

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

1998 Korea - Final Round, 2

Tags: inequalities , circumcircle , geometry

Let $D$,$E$,$F$ be points on the sides $BC$,$CA$,$AB$ respectively of a triangle $ABC$. Lines $AD$,$BE$,$CF$ intersect the circumcircle of $ABC$ again at $P$,$Q$,$R$, respectively.Show that: \[\frac{AD}{PD}+\frac{BE}{QE}+\frac{CF}{RF}\geq 9\] and find the cases of equality.