Olympiad problems

1992 Baltic Way, 19

Let $ C$ be a circle in plane. Let $ C_1$ and $ C_2$ be nonintersecting circles touching $ C$ internally at points $ A$ and $ B$ respectively. Let $ t$ be a common tangent of $ C_1$ and $ C_2$ touching them at points $ D$ and $ E$ respectively, such that both $ C_1$ and $ C_2$ are on the same side of $ t$. Let $ F$ be the point of intersection of $ AD$ and $ BE$. Show that $ F$ lies on $ C$.

Indonesia Regional MO OSP SMA - geometry, 2014.2

Tags: angle bisector , excenter , geometry

Given an acute triangle $ABC$ with $AB <AC$. The ex-circles of triangle $ABC$ opposite $B$ and $C$ are centered on $B_1$ and $C_1$, respectively. Let $D$ be the midpoint of $B_1C_1$. Suppose that $E$ is the point of intersection of $AB$ and $CD$, and $F$ is the point of intersection of $AC$ and $BD$. If $EF$ intersects $BC$ at point $G$, prove that $AG$ is the bisector of $\angle BAC$.

2020 Novosibirsk Oral Olympiad in Geometry, 6

Tags: angle bisector , angle , geometry

Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

2022 South Africa National Olympiad, 4

Tags: angle bisector , circumcircle , geometry

Let $ABC$ be a triangle with $AB < AC$. A point $P$ on the circumcircle of $ABC$ (on the same side of $BC$ as $A$) is chosen in such a way that $BP = CP$. Let $BP$ and the angle bisector of $\angle BAC$ intersect at $Q$, and let the line through $Q$ and parallel to $BC$ intersect $AC$ at $R$. Prove that $BR = CR$.

2007 All-Russian Olympiad, 4

Tags: cyclic quadrilateral , asymptote , circumcircle , geometry , angle bisector

$BB_{1}$ is a bisector of an acute triangle $ABC$. A perpendicular from $B_{1}$ to $BC$ meets a smaller arc $BC$ of a circumcircle of $ABC$ in a point $K$. A perpendicular from $B$ to $AK$ meets $AC$ in a point $L$. $BB_{1}$ meets arc $AC$ in $T$. Prove that $K$, $L$, $T$ are collinear. [i]V. Astakhov[/i]

2015 CentroAmerican, Problem 5

Tags: angle bisector , geometry

Let $ABC$ be a triangle such that $AC=2AB$. Let $D$ be the point of intersection of the angle bisector of the angle $CAB$ with $BC$. Let $F$ be the point of intersection of the line parallel to $AB$ passing through $C$ with the perpendicular line to $AD$ passing through $A$. Prove that $FD$ passes through the midpoint of $AC$.

2000 Junior Balkan Team Selection Tests - Moldova, 3

Tags: angle bisector , geometric inequality , isosceles , geometry

Let $ABC$ be a triangle with $AB = AC$ ¸ $\angle BAC = 100^o$ and $AD, BE$ angle bisectors. Prove that $2AD <BE + EA$

2000 239 Open Mathematical Olympiad, 7

Tags: circumcircle , euler , incenter , geometry , perpendicular bisector , angle bisector

The perpendicular bisectors of the sides AB and BC of a triangle ABC meet the lines BC and AB at the points X and Z, respectively. The angle bisectors of the angles XAC and ZCA intersect at a point B'. Similarly, define two points C' and A'. Prove that the points A', B', C' lie on one line through the incenter I of triangle ABC. [i]Extension:[/i] Prove that the points A', B', C' lie on the line OI, where O is the circumcenter and I is the incenter of triangle ABC. Darij

2000 May Olympiad, 2

Tags: geometry , midpoint , right triangle , angle bisector

Let $ABC$ be a right triangle in $A$ , whose leg measures $1$ cm. The bisector of the angle $BAC$ cuts the hypotenuse in $R$, the perpendicular to $AR$ on $R$ , cuts the side $AB$ at its midpoint. Find the measurement of the side $AB$ .

Indonesia MO Shortlist - geometry, g8

Tags: geometry , angle bisector , geometric inequality

Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are angle bisectors. Define $P_1$, $P_2$, $P_3$ respectively as the intersection point of $AD$ with $EF$, $BE$ with $DF$, $CF$ with $DE$ respectively. Prove that $$\frac{AD}{AP_1}+\frac{BE}{BP_2}+\frac{CF}{CP_3} \ge 6$$

2002 Silk Road, 1

Tags: circumcircle , angle bisector , geometry

Let $ \triangle ABC$ be a triangle with incircle $ \omega(I,r)$and circumcircle $ \zeta(O,R)$.Let $ l_{a}$ be the angle bisector of $ \angle BAC$.Denote $ P\equal{}l_{a}\cap\zeta$.Let $ D$ be the point of tangency $ \omega$ with $ [BC]$.Denote $ Q\equal{}PD\cap\zeta$.Show that $ PI\equal{}QI$ if $ PD\equal{}r$.

2014 India IMO Training Camp, 3

Tags: geometric transformation , trigonometry , angle bisector , function , circumcircle , geometry

In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.

2008 Mongolia Team Selection Test, 2

Tags: geometry , angle bisector , symmetry , incenter

The quadrilateral $ ABCD$ inscribed in a circle wich has diameter $ BD$. Let $ A',B'$ are symmetric to $ A,B$ with respect to the line $ BD$ and $ AC$ respectively. If $ A'C \cap BD \equal{} P$ and $ AC\cap B'D \equal{} Q$ then prove that $ PQ \perp AC$

2000 AIME Problems, 10

Tags: trapezoid , geometry , trigonometry , angle bisector

A circle is inscribed in quadrilateral $ABCD,$ tangent to $\overline{AB}$ at $P$ and to $\overline{CD}$ at $Q.$ Given that $AP=19, PB=26, CQ=37,$ and $QD=23,$ find the square of the radius of the circle.

2013 AMC 12/AHSME, 21

Tags: geometry , conic , combinatorics , angle bisector , parabola

Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form $y=ax+b$ with a and b integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas? ${ \textbf{(A)}\ 720\qquad\textbf{(B)}\ 760\qquad\textbf{(C)}\ 810\qquad\textbf{(D}}\ 840\qquad\textbf{(E)}\ 870 $

2008 Estonia Team Selection Test, 2

Tags: geometry , angle bisector , cyclic , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively. a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$. b) Does the converse implication also always hold?

2006 Estonia Math Open Junior Contests, 3

Tags: parallelogram , angle bisector , geometry

Let ABCD be a parallelogram, M the midpoint of AB and N the intersection of CD and the angle bisector of ABC. Prove that CM and BN are perpendicular iff AN is the angle bisector of DAB.

2013 ELMO Shortlist, 4

Tags: geometry , angle bisector , incenter , perpendicular bisector

Triangle $ABC$ is inscribed in circle $\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$. Prove that if $BE = BR$, then $\angle ELK = \tfrac{1}{2} \angle BCD$. [i]Proposed by Evan Chen[/i]

2005 Tournament of Towns, 4

Tags: square , geometry , angle bisector

$M$ and $N$ are the midpoints of sides $BC$ and $AD$, respectively, of a square $ABCD$. $K$ is an arbitrary point on the extension of the diagonal $AC$ beyond $A$. The segment $KM$ intersects the side $AB$ at some point $L$. Prove that $\angle KNA = \angle LNA$. [i](5 points)[/i]

2023 Bulgaria National Olympiad, 2

Tags: angle bisector , circumcircle , geometry

Let $ABC$ be an acute triangle and $A_{1}, B_{1}, C_{1}$ be the touchpoints of the excircles with the segments $BC, CA, AB$ respectively. Let $O_{A}, O_{B}, O_{C}$ be the circumcenters of $\triangle AB_{1}C_{1}, \triangle BC_{1}A_{1}, \triangle CA_{1}B_{1}$ respectively. Prove that the lines through $O_{A}, O_{B}, O_{C}$ respectively parallel to the internal angle bisectors of $\angle A,\angle B, \angle C$ are concurrent.

2018 Azerbaijan Junior NMO, 4

Tags: geometry , angle bisector , circles , tangent

A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$

2015 Moldova Team Selection Test, 2

Tags: incenter , geometry , angle bisector

Consider a triangle $\triangle ABC$, let the incircle centered at $I$ touch the sides $BC,CA,AB$ at points $D,E,F$ respectively. Let the angle bisector of $\angle BIC$ meet $BC$ at $M$, and the angle bisector of $\angle EDF$ meet $EF$ at $N$. Prove that $A,M,N$ are collinear.

2006 Hong Kong TST., 3

Tags: geometry , reflection , geometric transformation , angle bisector , trigonometry

In triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Find angle B.

1995 Tournament Of Towns, (466) 4

Tags: geometry , angle bisector , equal segments

From the vertex $A$ of a triangle $ABC$, three segments are drawn: the bisectors $AM$ and $AN$ of its interior and exterior angles and the tangent $AK$ to the circumscribed circle of the triangle (the points $M$, $K$ and $N$ lie on the line $BC$). Prove that $MK = KN$. (I Sharygin)

2019 Israel Olympic Revenge, P3

Tags: incenter , circumcircle , geometry , angle bisector

Let $ABCD$ be a circumscribed quadrilateral, assume $ABCD$ is not a kite. Denote the circumcenters of triangle $ABC,BCD,CDA,DAB$ by $O_D,O_A,O_B,O_C$ respectively. a. Prove that $O_AO_BO_CO_D$ is circumscribed. b. Let the angle bisector of $\angle BAD$ intersect the angle bisector of $\angle O_BO_AO_D$ in $X$. Similarly define the points $Y,Z,W$. Denote the incenters of $ABCD, O_AO_BO_CO_D$ by $I,J$ respectively. Express the angles $\angle ZYJ,\angle XYI$ in terms of angles of quadrilateral $ABCD$.