Olympiad problems

2011 Singapore Junior Math Olympiad, 2

Two circles $\Gamma_1, \Gamma_2$ with radii $r_i, r_2$, respectively, touch internally at the point $P$. A tangent parallel to the diameter through $P$ touches $ \Gamma_1$ at $R$ and intersects $\Gamma_2$ at $M$ and $N$. Prove that $PR$ bisects $\angle MPN$.

1999 Turkey Team Selection Test, 2

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

Let $L$ and $N$ be the mid-points of the diagonals $[AC]$ and $[BD]$ of the cyclic quadrilateral $ABCD$, respectively. If $BD$ is the bisector of the angle $ANC$, then prove that $AC$ is the bisector of the angle $BLD$.

2016 CCA Math Bonanza, T9

Tags: geometry , angle bisector

Let ABC be a triangle with $AB = 8$, $BC = 9$, and $CA = 10$. The line tangent to the circumcircle of $ABC$ at $A$ intersects the line $BC$ at $T$, and the circle centered at $T$ passing through $A$ intersects the line $AC$ for a second time at $S$. If the angle bisector of $\angle SBA$ intersects $SA$ at $P$, compute the length of segment $SP$. [i]2016 CCA Math Bonanza Team #9[/i]

2018 Costa Rica - Final Round, G2

Tags: geometry , equal segments , angle bisector

Consider $\vartriangle ABC$, with $AD$ bisecting $\angle BAC$, $D$ on segment $BC$. Let $E$ be a point on $BC$, such that $BD = EC$. Through $E$ we draw the line $\ell$ parallel to $AD$ and consider a point $ P$ on it and inside the $\vartriangle ABC$. Let $G$ be the point where line $BP$ cuts side $AC$ and let F be the point where line $CP$ to side $AB$. Show that $BF = CG$.

2018 CMIMC Geometry, 2

Tags: angle bisector , geometry

Let $ABCD$ be a square of side length $1$, and let $P$ be a variable point on $\overline{CD}$. Denote by $Q$ the intersection point of the angle bisector of $\angle APB$ with $\overline{AB}$. The set of possible locations for $Q$ as $P$ varies along $\overline{CD}$ is a line segment; what is the length of this segment?

2010 Iran MO (2nd Round), 5

Tags: geometry , circumcircle , trigonometry , vector , angle bisector

In triangle $ABC$ we havev $\angle A=\frac{\pi}{3}$. Construct $E$ and $F$ on continue of $AB$ and $AC$ respectively such that $BE=CF=BC$. Suppose that $EF$ meets circumcircle of $\triangle ACE$ in $K$. ($K\not \equiv E$). Prove that $K$ is on the bisector of $\angle A$.

1997 Tournament Of Towns, (560) 1

Tags: geometry , isosceles , equal segments , angle bisector

$M$ and $N$ are the midpoints of the sides $AB$ and $AC$ of a triangle ABC respectively. $P$ and $Q$ are points on the sides $AB$ and $AC$ respectively such that the bisector of the angle $ACB$ also bisects the angle $MCP$, and the bisector of the angle $ABC$ also bisects the angle $NBQ$. If $AP = AQ$, does it follow that $ABC$ is isosceles? (V Senderov)

2004 Vietnam National Olympiad, 2

Tags: angle bisector , geometry

In a triangle $ ABC$, the bisector of $ \angle ACB$ cuts the side $ AB$ at $ D$. An arbitrary circle $ (O)$ passing through $ C$ and $ D$ meets the lines $ BC$ and $ AC$ at $ M$ and $ N$ (different from $ C$), respectively. (a) Prove that there is a circle $ (S)$ touching $ DM$ at $ M$ and $ DN$ at $ N$. (b) If circle $ (S)$ intersects the lines $ BC$ and $ CA$ again at $ P$ and $ Q$ respectively, prove that the lengths of the segments $ MP$ and $ NQ$ are constant as $ (O)$ varies.

JBMO Geometry Collection, 2008

Tags: trigonometry , angle bisector , perpendicular bisector , power of a point , geometry

The vertices $ A$ and $ B$ of an equilateral triangle $ ABC$ lie on a circle $k$ of radius $1$, and the vertex $ C$ is in the interior of the circle $ k$. A point $ D$, different from $ B$, lies on $ k$ so that $ AD\equal{}AB$. The line $ DC$ intersects $ k$ for the second time at point $ E$. Find the length of the line segment $ CE$.

2014 India Regional Mathematical Olympiad, 6

Tags: incenter , trapezoid , angle bisector , geometry

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.

2004 Germany Team Selection Test, 2

Tags: geometry , angle bisector , menelaus

In a triangle $ABC$, let $D$ be the midpoint of the side $BC$, and let $E$ be a point on the side $AC$. The lines $BE$ and $AD$ meet at a point $F$. Prove: If $\frac{BF}{FE}=\frac{BC}{AB}+1$, then the line $BE$ bisects the angle $ABC$.

2007 Bosnia Herzegovina Team Selection Test, 1

Tags: angle bisector , geometry

Let $ABC$ be a triangle such that length of internal angle bisector from $B$ is equal to $s$. Also, length of external angle bisector from $B$ is equal to $s_1$. Find area of triangle $ABC$ if $\frac{AB}{BC} = k$

1995 Turkey Team Selection Test, 1

Tags: perpendicular bisector , angle bisector , geometry

In a convex quadrilateral $ABCD$ it is given that $\angle{CAB} = 40^{\circ}, \angle{CAD} = 30^{\circ}, \angle{DBA} = 75^{\circ}$, and $\angle{DBC}=25^{\circ}$. Find $\angle{BDC}$.

2016 Dutch IMO TST, 3

Tags: geometry , equal segments , isosceles , angle bisector

Let $\vartriangle ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $|BF| = |BE|$ and such that $ED$ is the internal angle bisector of $\angle BEC$. Prove that $|BD|= |EF|$ if and only if $|AF| = |EC|$.

Ukrainian From Tasks to Tasks - geometry, 2011.8

Tags: geometry , angle bisector , parallel , isosceles

On the median $AD$ of the isosceles triangle $ABC$, point $E$ is marked. Point $F$ is the projection of point $E$ on the line $BC$, point $M$ lies on the segment $EF$, points $N$ and $P$ are projections of point $M$ on the lines $AC$ and $AB$, respectively. Prove that the bisectors of the angles $PMN$ and $PEN$ are parallel.

2000 China Team Selection Test, 1

Tags: circumcircle , ellipse , angle bisector , geometry , conic

Let $ABC$ be a triangle such that $AB = AC$. Let $D,E$ be points on $AB,AC$ respectively such that $DE = AC$. Let $DE$ meet the circumcircle of triangle $ABC$ at point $T$. Let $P$ be a point on $AT$. Prove that $PD + PE = AT$ if and only if $P$ lies on the circumcircle of triangle $ADE$.

2006 Iran Team Selection Test, 5

Tags: circumcircle , trigonometry , parallelogram , angle bisector , geometry

Let $ABC$ be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to $A$. Suppose that the outer inscribed circle with respect to $A$ touches $BC,AC,AB$ at $M,N,L$. Prove that $O$ (Center of circumcircle) is the orthocenter of $MNL$.

2006 France Team Selection Test, 1

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

Let $ABCD$ be a square and let $\Gamma$ be the circumcircle of $ABCD$. $M$ is a point of $\Gamma$ belonging to the arc $CD$ which doesn't contain $A$. $P$ and $R$ are respectively the intersection points of $(AM)$ with $[BD]$ and $[CD]$, $Q$ and $S$ are respectively the intersection points of $(BM)$ with $[AC]$ and $[DC]$. Prove that $(PS)$ and $(QR)$ are perpendicular.

2012 ITAMO, 1

Tags: angle bisector , similar triangles , geometry

On the sides of a triangle $ABC$ right angled at $A$ three points $D, E$ and $F$ (respectively $BC, AC$ and $AB$) are chosen so that the quadrilateral $AFDE$ is a square. If $x$ is the length of the side of the square, show that \[\frac{1}{x}=\frac{1}{AB}+\frac{1}{AC}\]

2003 Iran MO (3rd Round), 4

Tags: vector , angle bisector , geometry

XOY is angle in the plane.A,B are variable point on OX,OY such that 1/OA+1/OB=1/K (k is constant).draw two circles with diameter OA and OB.prove that common external tangent to these circles is tangent to the constant circle( ditermine the radius and the locus of its center).

2000 All-Russian Olympiad, 7

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

A quadrilateral $ABCD$ is circumscribed about a circle $\omega$. The lines $AB$ and $CD$ meet at $O$. A circle $\omega_1$ is tangent to side $BC$ at $K$ and to the extensions of sides $AB$ and $CD$, and a circle $\omega_2$ is tangent to side $AD$ at $L$ and to the extensions of sides $AB$ and $CD$. Suppose that points $O$, $K$, $L$ lie on a line. Prove that the midpoints of $BC$ and $AD$ and the center of $\omega$ also lie on a line.

2017-IMOC, G1

Tags: geometry , angle bisector

Given $\vartriangle ABC$. Choose two points $P, Q$ on $AB, AC$ such that $BP = CQ$. Let $M, T$ be the midpoints of $BC, PQ$. Show that $MT$ is parallel to the angle bisevtor of $\angle BAC$ [img]http://4.bp.blogspot.com/-MgMtdnPtq1c/XnSHHFl1LDI/AAAAAAAALdY/8g8541DnyGo_Gqd19-7bMBpVRFhbXeYPACK4BGAYYCw/s1600/imoc2017%2Bg1.png[/img]

2013 ELMO Shortlist, 4

Tags: geometry , incenter , angle bisector , 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]

2016 Thailand Mathematical Olympiad, 1

Tags: geometry , perpendicular bisector , ratio , angle bisector

Let $ABC$ be a triangle with $AB \ne AC$. Let the angle bisector of $\angle BAC$ intersects $BC$ at $P$ and intersects the perpendicular bisector of segment $BC$ at $Q$. Prove that $\frac{PQ}{AQ} =\left( \frac{BC}{AB + AC}\right)^2$

2008 Estonia Team Selection Test, 2

Tags: geometry , cyclic , cyclic quadrilateral , angle bisector

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?