Olympiad problems

2018 Dutch IMO TST, 4

Tags: geometry , circumcircle , incenter , tangent , arc midpoint

In a non-isosceles triangle $ABC$ the centre of the incircle is denoted by $I$. The other intersection point of the angle bisector of $\angle BAC$ and the circumcircle of $\vartriangle ABC$ is $D$. The line through $I$ perpendicular to $AD$ intersects $BC$ in $F$. The midpoint of the circle arc $BC$ on which $A$ lies, is denoted by $M$. The other intersection point of the line $MI$ and the circle through $B, I$ and $C$, is denoted by $N$. Prove that $FN$ is tangent to the circle through $B, I$ and $C$.

2003 Olympic Revenge, 1

Tags: perpendicular , geometry , circumcircle , arc midpoint , isosceles

Let $ABC$ be a triangle with circumcircle $\Gamma$. $D$ is the midpoint of arc $BC$ (this arc does not contain $A$). $E$ is the common point of $BC$ and the perpendicular bisector of $BD$. $F$ is the common point of $AC$ and the parallel to $AB$ containing $D$. $G$ is the common point of $EF$ and $AB$. $H$ is the common point of $GD$ and $AC$. Show that $GAH$ is isosceles.

2018 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: circumcircle , geometry , midpoint , arc midpoint , intersection

Let $ABC$ be an acute triangle with $AC> AB$. be $\Gamma$ the circumcircle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smallest arc $BC$ of this circle. Let $E$ and $F$ points of the segments $AB$ and $AC$ respectively such that $AE = AF$. Let $P \neq A$ be the second intersection point of the circumcircle circumscribed to $AEF$ with $\Gamma$. Let $G$ and $H$ be the intersections of lines $PE$ and $PF$ with $\Gamma$ other than $P$, respectively. Let $J$ and $K$ be the intersection points of lines $DG$ and $DH$ with lines $AB$ and $AC$ respectively. Show that the $JK$ line passes through the midpoint of $BC$

2007 Bosnia and Herzegovina Junior BMO TST, 4

Tags: geometry , circumcircle , incenter , arc midpoint , equal angles

Let $I$ be the incenter of the triangle $ABC$ ($AB < BC$). Let $M$ be the midpoint of $AC$, and let $N$ be the midpoint of the arc $AC$ of the circumcircle of $ABC$ which contains $B$. Prove that $\angle IMA = \angle INB$.

2018 Oral Moscow Geometry Olympiad, 4

Tags: geometry , arc midpoint , construction

Given a triangle $ABC$ ($AB> AC$) and a circle circumscribed around it. Construct with a compass and a ruler the midpoint of the arc $BC$ (not containing vertex $A$), with no more than two lines (straight or circles).

VMEO III 2006, 10.1

Tags: geometry , concurrency , arc midpoint

Let $ABC$ be a triangle inscribed in a circle with center $O$. Let $A_1$ be a point on arc $BC$ that does not contain $ A$ such that the line perpendicular to $OA$ at $A_1$ intersects the lines $AB$ and $AC$ at two points and the line segment joining those two points has as midpoint $A_1$. Points $B_1$, $C_1$ are determined similarly. Prove that the lines $AA_1$, $BB_1$, $CC_1$ are concurrent.

2013 Oral Moscow Geometry Olympiad, 6

Tags: trapezoid , incircle , arc midpoint , concurrency , geometry

The trapezoid $ABCD$ is inscribed in the circle $w$ ($AD // BC$). The circles inscribed in the triangles $ABC$ and $ABD$ touch the base of the trapezoid $BC$ and $AD$ at points $P$ and $Q$ respectively. Points $X$ and $Y$ are the midpoints of the arcs $BC$ and $AD$ of circle $w$ that do not contain points $A$ and $B$ respectively. Prove that lines $XP$ and $YQ$ intersect on the circle $w$.

2023 Germany Team Selection Test, 3

Tags: geometry , cyclic quadrilateral , arc midpoint

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

2019 Tuymaada Olympiad, 8

Tags: geometric transformation , circumcircle , geometry , arc midpoint , moving points , inversion , reflection

In $\triangle ABC$ $\angle B$ is obtuse and $AB \ne BC$. Let $O$ is the circumcenter and $\omega$ is the circumcircle of this triangle. $N$ is the midpoint of arc $ABC$. The circumcircle of $\triangle BON$ intersects $AC$ on points $X$ and $Y$. Let $BX \cap \omega = P \ne B$ and $BY \cap \omega = Q \ne B$. Prove that $P, Q$ and reflection of $N$ with respect to line $AC$ are collinear.

2023 Estonia Team Selection Test, 3

Tags: cyclic quadrilateral , geometry , arc midpoint

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

2019 Saint Petersburg Mathematical Olympiad, 3

Tags: midpoint , inequalities , geometric inequality , arc midpoint

Prove that the distance between the midpoint of side $BC$ of triangle $ABC$ and the midpoint of the arc $ABC$ of its circumscribed circle is not less than $AB / 2$

2022 IMO Shortlist, G2

Tags: cyclic quadrilateral , geometry , arc midpoint

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

1998 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: geometry , fixed point , arc midpoint , arc , circles

Consider an arc $AB$ of a circle $C$ and a point $P$ variable in that arc $AB$. Let $D$ be the midpoint of the arc $AP$ that doeas not contain $B$ and let $E$ be the midpoint of the arc $BP$ that does not contain $A$. Let $C_1$ be the circle with center $D$ passing through $A$ and $C_2$ be the circle with center $E$ passing through $B.$ Prove that the line that contains the intersection points of $C_1$ and $C_2$ passes through a fixed point.

2021 239 Open Mathematical Olympiad, 1

Tags: angle bisector , geometry , circumcircle , angle , arc midpoint

Points $X$ and $Y$ are the midpoints of arcs $AB$ and $BC$ of the circumscribed circle of triangle $ABC$. Point $T$ lies on side $AC$. It turned out that the bisectors of the angles $ATB$ and $BTC$ pass through points $X$ and $Y$ respectively. What angle $B$ can be in triangle $ABC$?

2018 OMMock - Mexico National Olympiad Mock Exam, 5

Tags: circumcircle , geometry , arc midpoint

Let $ABC$ be a triangle with circumcirle $\Gamma$, and let $M$ and $N$ be the respective midpoints of the minor arcs $AB$ and $AC$ of $\Gamma$. Let $P$ and $Q$ be points such that $AB=BP$, $AC=CQ$, and $P$, $B$, $C$, $Q$ lie on $BC$ in that order. Prove that $PM$ and $QN$ meet at a point on $\Gamma$. [i]Proposed by Victor Domínguez[/i]

2021 OMpD, 2

Tags: geometry , perpendicular , arc midpoint

Let $ABC$ be a triangle, $\Gamma$ its circumcircle and $D$ the midpoint of the arc $AC$ of $\Gamma$ that does not contain $B$. If $O$ is the center of $\Gamma$ and I is the incenter of $ABC$, prove that $OI$ is perpendicular to $BD$ if and only if $AB + BC = 2AC$.

2000 Switzerland Team Selection Test, 1

Tags: geometry , arc midpoint , perpendicular

A convex quadrilateral $ABCD$ is inscribed in a circle. Show that the line connecting the midpoints of the arcs $AB$ and $CD$ and the line connecting the midpoints of the arcs $BC$ and $DA$ are perpendicular.

2021 Yasinsky Geometry Olympiad, 6

Tags: geometry , incenter , arc midpoint , right angle

In an acute-angled triangle $ABC$, point $I$ is the center of the inscribed circle, point $T$ is the midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$. It turned out that $\angle AIT = 90^o$ . Prove that $AB + AC = 3BC$. (Matthew of Kursk)