Olympiad problems

2002 National Olympiad First Round, 9

Let $ABC$ be triangle such that $|AB| = 5$, $|BC| = 9$ and $|AC| = 8$. The angle bisector of $\widehat{BCA}$ meets $BA$ at $X$ and the angle bisector of $\widehat{CAB}$ meets $BC$ at $Y$. Let $Z$ be the intersection of lines $XY$ and $AC$. What is $|AZ|$? $ \textbf{a)}\ \sqrt{104} \qquad\textbf{b)}\ \sqrt{145} \qquad\textbf{c)}\ \sqrt{89} \qquad\textbf{d)}\ 9 \qquad\textbf{e)}\ 10 $

Swiss NMO - geometry, 2015.1

Tags: circumcircle , geometry , angle bisector , ratio

Let $ABC$ be an acute-angled triangle with $AB \ne BC$ and radius $k$. Let $P$ and $Q$ be the points of intersection of $k$ with the internal bisector and the external bisector of $\angle CBA$ respectively. Let $D$ be the intersection of $AC$ and $PQ$. Find the ratio $AD: DC$.

2014 Junior Regional Olympiad - FBH, 3

Tags: geometry , angle bisector

Let $ABC$ be a right angled triangle. Prove that angle bisector of right angle is simultaneously an angle bisector of angle between median and altitude to hypotenuse.

2003 JBMO Shortlist, 7

Tags: geometry , angle bisector , circumcircle

Let $D$, $E$, $F$ be the midpoints of the arcs $BC$, $CA$, $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$, $B$, $C$, respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$, and let $M$ be the midpoint of the segment $GH$. Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$, and let $N$ be the midpoint of the segment $KJ$. a) Find the angles of triangle $DMN$; b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$, then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$.

2000 Belarus Team Selection Test, 5.1

Tags: inequalities , angle bisector , geometric inequalities , median

Let $AM$ and $AL$ be the median and bisector of a triangle $ABC$ ($M,L \in BC$). If $BC = a, AM = m_a, AL = l_a$, prove the inequalities: (a) $a\tan \frac{a}{2} \le 2m_a \le a \cot \frac{a}{2} $ if $a < \frac{\pi}{2}$ and $a\tan \frac{a}{2} \ge 2m_a \ge a \cot \frac{a}{2} $ if $a > \frac{\pi}{2}$ (b) $2l_a \le a\cot \frac{a}{2} $.

2021 Korea National Olympiad, P1

Tags: geometry , circumcircle , angle bisector

Let $ABC$ be an acute triangle and $D$ be an intersection of the angle bisector of $A$ and side $BC$. Let $\Omega$ be a circle tangent to the circumcircle of triangle $ABC$ and side $BC$ at $A$ and $D$, respectively. $\Omega$ meets the sides $AB, AC$ again at $E, F$, respectively. The perpendicular line to $AD$, passing through $E, F$ meets $\Omega$ again at $G, H$, respectively. Suppose that $AE$ and $GD$ meet at $P$, $EH$ and $GF$ meet at $Q$, and $HD$ and $AF$ meet at $R$. Prove that $\dfrac{\overline{QF}}{\overline{QG}}=\dfrac{\overline{HR}}{\overline{PG}}$.

2007 CentroAmerican, 2

Tags: symmetry , angle bisector , geometry

In a triangle $ABC$, the angle bisector of $A$ and the cevians $BD$ and $CE$ concur at a point $P$ inside the triangle. Show that the quadrilateral $ADPE$ has an incircle if and only if $AB=AC$.

2016 Bangladesh Mathematical Olympiad, 6

Tags: geometry , angle bisector , parallel

$\triangle ABC$ is an isosceles triangle with $AC = BC$ and $\angle ACB < 60^{\circ}$. $I$ and $O$ are the incenter and circumcenter of $\triangle ABC$. The circumcircle of $\triangle BIO$ intersects $BC$ at $D \neq B$. (a) Do the lines $AC$ and $DI$ intersect? Give a proof. (b) What is the angle of intersection between the lines $OD$ and $IB$?

2025 Israel TST, P3

Tags: circumcircle , cyclic quadrilateral , angle bisector , geometry

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. The internal angle bisectors of $\angle DAB$, $\angle ABC$, $\angle BCD$, $\angle CDA$ create a convex quadrilateral $Q_1$. The external bisectors of the same angles create another convex quadrilateral $Q_2$. Prove $Q_1$, $Q_2$ are cyclic, and that $O$ is the midpoint of their circumcenters.

2024 JHMT HS, 8

Tags: angle bisector , circumcircle , geometry

Points $A$, $B$, $C$, and $D$ lie on a circle $\Gamma$, in that order, with $AB=5$ and $AD=3$. The angle bisector of $\angle ABC$ intersects $\Gamma$ at point $E$ on the opposite side of $\overleftrightarrow{CD}$ as $A$ and $B$. Assume that $\overline{BE}$ is a diameter of $\Gamma$ and $AC=AE$. Compute $DE$.

2004 CentroAmerican, 2

Tags: angle bisector , trapezoid , geometry , circumcircle

Let $ABCD$ be a trapezium such that $AB||CD$ and $AB+CD=AD$. Let $P$ be the point on $AD$ such that $AP=AB$ and $PD=CD$. $a)$ Prove that $\angle BPC=90^{\circ}$. $b)$ $Q$ is the midpoint of $BC$ and $R$ is the point of intersection between the line $AD$ and the circle passing through the points $B,A$ and $Q$. Show that the points $B,P,R$ and $C$ are concyclic.

2007 Junior Balkan MO, 2

Tags: trigonometry , geometry , circumcircle , incenter , angle bisector , cyclic quadrilateral , function

Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.

2003 Junior Tuymaada Olympiad, 7

Tags: geometry , perpendicular , angle bisector , tangent

Through the point $ K $ lying outside the circle $ \omega $, the tangents are drawn $ KB $ and $ KD $ to this circle ($ B $ and $ D $ are tangency points) and a line intersecting a circle at points $ A $ and $ C $. The bisector of angle $ ABC $ intersects the segment $ AC $ at the point $ E $ and circle $ \omega $ at $ F $. Prove that $ \angle FDE = 90^\circ $.

1981 Poland - Second Round, 2

Tags: angle bisector , circles , geometry

Two circles touch internally at point $P$. A line tangent to one of the circles at point $A$ intersects the other circle at points $B$ and $C$. Prove that the line $ PA $ is the bisector of the angle $ BPC $.

2008 Hungary-Israel Binational, 3

Tags: reflection , geometry , angle bisector , conic , hyperbola , geometric transformation

P and Q are 2 points in the area bounded by 2 rays, e and f, coming out from a point O. Describe how to construct, with a ruler and a compass only, an isosceles triangle ABC, such that his base AB is on the ray e, the point C is on the ray f, P is on AC, and Q on BC.

Kyiv City MO Seniors 2003+ geometry, 2003.10.4

Tags: angle bisector , geometry

Let $ABCD$ be a convex quadrilateral. The bisector of the angle $ACD$ intersects $BD$ at point $E$. It is known that $\angle CAD = \angle BCE= 90^o$. Prove that the $AC$ is the bisector of the angle $BAE$ . (Nikolay Nikolay)

2004 Baltic Way, 19

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

Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$. Let $M$ be a point on the side $BC$ such that $\angle BAM = \angle DAC$. Further, let $L$ be the second intersection point of the circumcircle of the triangle $CAM$ with the side $AB$, and let $K$ be the second intersection point of the circumcircle of the triangle $BAM$ with the side $AC$. Prove that $KL \parallel BC$.

2021 Yasinsky Geometry Olympiad, 4

Tags: geometry , orthocenter , angle bisector

In triangle $ABC$, the point $H$ is the orthocenter. A circle centered at point $H$ and with radius $AH$ intersects the lines $AB$ and $AC$ at points $E$ and $D$, respectively. The point $X$ is the symmetric of the point $A$ with respect to the line $BC$ . Prove that $XH$ is the bisector of the angle $DXE$. (Matthew of Kursk)

2014 India Regional Mathematical Olympiad, 6

Tags: geometry , angle bisector , trapezoid , incenter

Let $D,E,F$ be the points of contact of the incircle of an acute-angled triangle $ABC$ with $BC,CA,AB$ respectively. Let $I_1,I_2,I_3$ be the incentres of the triangles $AFE, BDF, CED$, respectively. Prove that the lines $I_1D, I_2E, I_3F$ are concurrent.

2014 Purple Comet Problems, 20

Tags: geometry , angle bisector

Triangle $ABC$ has a right angle at $C$. Let $D$ be the midpoint of side $\overline{AC}$, and let $E$ be the intersection of $\overline{AC}$ and the bisector of $\angle ABC$. The area of $\triangle ABC$ is $144$, and the area of $\triangle DBE$ is $8$. Find $AB^2$.

Indonesia MO Shortlist - geometry, g7

Tags: geometry , angle bisector

Given a convex quadrilateral $ABCD$, such that $OA = \frac{OB \cdot OD}{OC + CD}$ where $O$ is the intersection of the two diagonals. The circumcircle of triangle $ABC$ intersects $BD$ at point $Q$. Prove that $CQ$ bisects $\angle ACD$

2010 Thailand Mathematical Olympiad, 3

Tags: angle bisector , geometry , projection

Let $\vartriangle ABC$ be a scalene triangle with $AB < BC < CA$. Let $D$ be the projection of $A$ onto the angle bisector of $\angle ABC$, and let $E$ be the projection of $A$ onto the angle bisector of $\angle ACB$. The line $DE$ cuts sides $AB$ and AC at $M$ and $N$, respectively. Prove that $$\frac{AB+AC}{BC} =\frac{DE}{MN} + 1$$

2005 District Olympiad, 2

Tags: geometry , reflection , angle bisector , geometric transformation

Let $ABC$ be a triangle and let $M$ be the midpoint of the side $AB$. Let $BD$ be the interior angle bisector of $\angle ABC$, $D\in AC$. Prove that if $MD \perp BD$ then $AB=3BC$.

V Soros Olympiad 1998 - 99 (Russia), 10.6

Tags: geometry , angle bisector , angle

In triangle $ABC$, the bisectors of the internal angles $AA_1$ , $BB_1$ and $CC_1$ are drawn ($A_1, B_1$, $C_1$ - on the sides of the triangle). It is known that $\angle AA_1C = \angle AC_1B_1$. Find $\angle BCA$.

2008 Iran Team Selection Test, 12

Tags: symmetry , reflection , circumcircle , angle bisector , incenter , geometric transformation , geometry

In the acute-angled triangle $ ABC$, $ D$ is the intersection of the altitude passing through $ A$ with $ BC$ and $ I_a$ is the excenter of the triangle with respect to $ A$. $ K$ is a point on the extension of $ AB$ from $ B$, for which $ \angle AKI_a\equal{}90^\circ\plus{}\frac 34\angle C$. $ I_aK$ intersects the extension of $ AD$ at $ L$. Prove that $ DI_a$ bisects the angle $ \angle AI_aB$ iff $ AL\equal{}2R$. ($ R$ is the circumradius of $ ABC$)