Olympiad problems

2009 Romania Team Selection Test, 1

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$

2024 Moldova EGMO TST, 5

Tags: geometry , angle bisector , circumcircle , geometric inequality , inequalities

$AD$ Is the angle bisector Of $\angle BAC$ Where $D$ lies on the The circumcircle of $\triangle ABC$. Show that $2AD>AB+AC$

2006 Iran MO (2nd round), 2

Tags: circumcircle , angle bisector , geometry

Let $ABCD$ be a convex cyclic quadrilateral. Prove that: $a)$ the number of points on the circumcircle of $ABCD$, like $M$, such that $\frac{MA}{MB}=\frac{MD}{MC}$ is $4$. $b)$ The diagonals of the quadrilateral which is made with these points are perpendicular to each other.

1974 Poland - Second Round, 3

Tags: 3d geometry , tetrahedron , angle bisector , geometry

Prove that the orthogonal projections of the vertex $ D $ of the tetrahedron $ ABCD $ onto the bisectors of the internal and external dihedral angles at the edges $ \overline{AB} $, $ \overline{BC} $ and $ \overline{CA} $ belong to one plane .

2011 Bosnia and Herzegovina Junior BMO TST, 3

Tags: geometry , isosceles , perpendicular , angle bisector , ratio

In isosceles triangle $ABC$ ($AC=BC$), angle bisector $\angle BAC$ and altitude $CD$ from point $C$ intersect at point $O$, such that $CO=3 \cdot OD$. In which ratio does altitude from point $A$ on side $BC$ divide altitude $CD$ of triangle $ABC$

2024 Ecuador NMO (OMEC), 3

Tags: geometry , angle bisector , rectangle

Let $\triangle ABC$ with $\angle BAC=120 ^\circ$. Let $D, E, F$ points on sides $BC, CA, AB$, respectively, such that $AD, BE, CF$ are angle bisectors on $\triangle ABC$. \\ Prove that $\triangle ABC$ is isosceles if and only if $\triangle DEF$ is right-angled isosceles.

2004 Germany Team Selection Test, 1

Tags: parallelogram , trigonometry , angle bisector , geometry

The $A$-excircle of a triangle $ABC$ touches the side $BC$ at the point $K$ and the extended side $AB$ at the point $L$. The $B$-excircle touches the lines $BA$ and $BC$ at the points $M$ and $N$, respectively. The lines $KL$ and $MN$ meet at the point $X$. Show that the line $CX$ bisects the angle $ACN$.

2022 Yasinsky Geometry Olympiad, 2

Tags: angle bisector , bisector , angle , geometry

In the triangle $ABC$, angle $C$ is four times smaller than each of the other two angle The altitude $AK$ and the angle bisector $AL$ are drawn from the vertex of the angle $A$. It is known that the length of $AL$ is equal to $\ell$. Find the length of the segment $LK$. (Gryhoriy Filippovskyi)

2018 Mexico National Olympiad, 1

Tags: inequalities , geometry , angle bisector

Let $A$ and $B$ be two points on a line $\ell$, $M$ the midpoint of $AB$, and $X$ a point on segment $AB$ other than $M$. Let $\Omega$ be a semicircle with diameter $AB$. Consider a point $P$ on $\Omega$ and let $\Gamma$ be the circle through $P$ and $X$ that is tangent to $AB$. Let $Q$ be the second intersection point of $\Omega$ and $\Gamma$. The internal angle bisector of $\angle PXQ$ intersects $\Gamma$ at a point $R$. Let $Y$ be a point on $\ell$ such that $RY$ is perpendicular to $\ell$. Show that $MX > XY$

2008 Saint Petersburg Mathematical Olympiad, 3

Tags: angle bisector , geometry

Pentagon $ABCDE$ has circle $S$ inscribed into it. Side $BC$ is tangent to $S$ at point $K$. If $AB=BC=CD$, prove that angle $EKB$ is a right angle.

2001 Tournament Of Towns, 3

Tags: geometric transformation , reflection , angle bisector , geometry

Point $A$ lies inside an angle with vertex $M$. A ray issuing from point $A$ is reflected in one side of the angle at point $B$, then in the other side at point $C$ and then returns back to point $A$ (the ordinary rule of reflection holds). Prove that the center of the circle circumscribed about triangle $\triangle BCM$ lies on line $AM$.

2022 Iran MO (3rd Round), 1

Tags: geometry , angle chasing , spiral similarity , circumcircle , geometric transformation , reflection , angle bisector

Triangle $ABC$ is assumed. The point $T$ is the second intersection of the symmedian of vertex $A$ with the circumcircle of the triangle $ABC$ and the point $D \neq A$ lies on the line $AC$ such that $BA=BD$. The line that at $D$ tangents to the circumcircle of the triangle $ADT$, intersects the circumcircle of the triangle $DCT$ for the second time at $K$. Prove that $\angle BKC = 90^{\circ}$(The symmedian of the vertex $A$, is the reflection of the median of the vertex $A$ through the angle bisector of this vertex).

Novosibirsk Oral Geo Oly VIII, 2021.4

Tags: geometry , angle bisector , angle

Angle bisectors $AD$ and $BE$ are drawn in triangle $ABC$. It turned out that $DE$ is the bisector of triangle $ADC$. Find the angle $BAC$.

2011 Olympic Revenge, 4

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

Let $ABCD$ to be a quadrilateral inscribed in a circle $\Gamma$. Let $r$ and $s$ to be the tangents to $\Gamma$ through $B$ and $C$, respectively, $M$ the intersection between the lines $r$ and $AD$ and $N$ the intersection between the lines $s$ and $AD$. After all, let $E$ to be the intersection between the lines $BN$ and $CM$, $F$ the intersection between the lines $AE$ and $BC$ and $L$ the midpoint of $BC$. Prove that the circuncircle of the triangle $DLF$ is tangent to $\Gamma$.

2006 Purple Comet Problems, 21

Tags: geometry , angle bisector

In triangle $ABC$, $AB = 52$, $BC = 56$, $CA = 60$. Let $D$ be the foot of the altitude from $A$ and $E$ be the intersection of the internal angle bisector of $\angle BAC$ with $BC$. Find $DE$.

2010 Flanders Math Olympiad, 3

Tags: geometry , angle bisector , parallel , perpendicular , angle

In a triangle $ABC$, $\angle B= 2\angle A \ne 90^o$ . The inner bisector of $B$ intersects the perpendicular bisector of $[AC]$ at a point $D$. Prove that $AB \parallel CD$.

1998 Turkey MO (2nd round), 2

Tags: geometric transformation , reflection , parallelogram , angle bisector , hjelmslev , geometry

Variable points $M$ and $N$ are considered on the arms $\left[ OX \right.$ and $\left[ OY \right.$ , respectively, of an angle $XOY$ so that $\left| OM \right|+\left| ON \right|$ is constant. Determine the locus of the midpoint of $\left[ MN \right]$.

2005 ITAMO, 3

Tags: circumcircle , incenter , greatest common divisor , angle bisector , geometry

Two circles $\gamma_1, \gamma_2$ in a plane, with centers $A$ and $B$ respectively, intersect at $C$ and $D$. Suppose that the circumcircle of $ABC$ intersects $\gamma_1$ in $E$ and $\gamma_2$ in $F$, where the arc $EF$ not containing $C$ lies outside $\gamma_1$ and $\gamma_2$. Prove that this arc $EF$ is bisected by the line $CD$.

2019 India PRMO, 10

Tags: angle bisector , circumcircle , geometry

Let $ABC$ be a triangle and let $\Omega$ be its circumcircle. The internal bisectors of angles $A, B$ and $C$ intersect $\Omega$ at $A_1, B_1$ and $C_1$, respectively, and the internal bisectors of angles $A_1, B_1$ and $C_1$ of the triangles $A_1 A_2 A_ 3$ intersect $\Omega$ at $A_2, B_2$ and $C_2$, respectively. If the smallest angle of the triangle $ABC$ is $40^{\circ}$, what is the magnitude of the smallest angle of the triangle $A_2 B_2 C_2$ in degrees?

Champions Tournament Seniors - geometry, 2003.1

Tags: concurrency , perpendicular , angle bisector , geometry

Consider the triangle $ABC$, in which $AB > AC$. Let $P$ and $Q$ be the feet of the perpendiculars dropped from the vertices $B$ and $C$ on the bisector of the angle $BAC$, respectively. On the line $BC$ note point $B$ such that $AD \perp AP.$ Prove that the lines $BQ, PC$ and $AD$ intersect at one point.

Geometry Mathley 2011-12, 14.2

Tags: angle bisector , geometry , euler , euler circle , tangent circle

The nine-point Euler circle of triangle $ABC$ is tangent to the excircles in the angle $A,B,C$ at $Fa, Fb, Fc$ respectively. Prove that $AF_a$ bisects the angle $\angle CAB$ if and only if $AFa$ bisects the angle $\angle F_bAF_c$. Đỗ Thanh Sơn

2014 Abels Math Contest (Norwegian MO) Final, 2

Tags: geometry , angle bisector , concurrency , parallelogram

The points $P$ and $Q$ lie on the sides $BC$ and $CD$ of the parallelogram $ABCD$ so that $BP = QD$. Show that the intersection point between the lines $BQ$ and $DP$ lies on the line bisecting $\angle BAD$.

2008 National Olympiad First Round, 21

Tags: angle bisector , geometry

Let $ABC$ be a right triangle with $m(\widehat{A})=90^\circ$. Let $APQR$ be a square with area $9$ such that $P\in [AC]$, $Q\in [BC]$, $R\in [AB]$. Let $KLMN$ be a square with area $8$ such that $N,K\in [BC]$, $M\in [AB]$, and $L\in [AC]$. What is $|AB|+|AC|$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 16 $

2023 4th Memorial "Aleksandar Blazhevski-Cane", P4

Tags: cyclic quadrilateral , segment sum , angle bisector , circumcenter , circumcircle , geometry

Let $ABCD$ be a cyclic quadrilateral such that $AB = AD + BC$ and $CD < AB$. The diagonals $AC$ and $BD$ intersect at $P$, while the lines $AD$ and $BC$ intersect at $Q$. The angle bisector of $\angle APB$ meets $AB$ at $T$. Show that the circumcenter of the triangle $CTD$ lies on the circumcircle of the triangle $CQD$. [i]Proposed by Nikola Velov[/i]

2010 India IMO Training Camp, 7

Tags: geometry , circumcircle , geometric transformation , dilation , reflection , parallelogram , angle bisector

Let $ABCD$ be a cyclic quadrilaterla and let $E$ be the point of intersection of its diagonals $AC$ and $BD$. Suppose $AD$ and $BC$ meet in $F$. Let the midpoints of $AB$ and $CD$ be $G$ and $H$ respectively. If $\Gamma $ is the circumcircle of triangle $EGH$, prove that $FE$ is tangent to $\Gamma $.