Olympiad problems

2013 Kyiv Mathematical Festival, 3

Let $ABCD$ be a parallelogram ($AB < BC$). The bisector of the angle $BAD$ intersects the side $BC$ at the point K; and the bisector of the angle $ADC$ intersects the diagonal $AC$ at the point $F$. Suppose that $KD \perp BC$. Prove that $KF \perp BD$.

2020 Macedonian Nationаl Olympiad, 3

Tags: geometry , Ugly , Angle Bisectors , angle bisector , circumcircle

Let $ABC$ be a triangle, and $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$, respectively, such that $AA_1, BB_1, CC_1$ are the internal angle bisectors of $\triangle ABC$. The circumcircle $k' = (A_1B_1C_1)$ touches the side $BC$ at $A_1$. Let $B_2$ and $C_2$, respectively, be the second intersection points of $k'$ with lines $AC$ and $AB$. Prove that $|AB| = |AC|$ or $|AC_1| = |AB_2|$.

2022 Yasinsky Geometry Olympiad, 4

Tags: geometry , Angle Bisectors

Let $X$ be an arbitrary point on side $BC$ of triangle $ABC$. Triangle $T$ is formed by the angle bisectors of the angles $\angle ABC$, $\angle ACB$ and $\angle AXC$. Prove that the circle circumscribed around the triangle $T$, passes through the vertex $A$. (Dmytro Prokopenko)

Croatia MO (HMO) - geometry, 2013.7

Tags: geometry , angles , angle bisector , Angle Bisectors

In triangle $ABC$, the angle at vertex $B$ is $120^o$. Let $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$ respectively such that $AA_1, BB_1, CC_1$ are bisectors of the angles of triangle $ABC$. Determine the angle $\angle A_1B_1C_1$.

Indonesia MO Shortlist - geometry, g8

Tags: geometry , geometric inequality , Angle Bisectors , angle bisector

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 $$\frac{AD}{AP_1}+\frac{BE}{BP_2}+\frac{CF}{CP_3} \ge 6$$

2011 Portugal MO, 5

Tags: geometry , incircle , projections , Angle Bisectors

Let $[ABC]$ be a triangle, $D$ be the orthogonal projection of $B$ on the bisector of $\angle ACB$ and $E$ the orthogonal projection of $C$ on the bisector of $\angle ABC$ . Prove that $DE$ intersects the sides $[AB]$ and $[AC]$ at the touchpoints of the circle inscribed in the triangle $[ABC]$.

1999 Mexico National Olympiad, 5

Tags: geometry , Angle Bisectors

In a quadrilateral $ABCD$ with $AB // CD$, the external bisectors of the angles at $B$ and $C$ meet at $P$, while the external bisectors of the angles at $A$ and $D$ meet at $Q$. Prove that the length of $PQ$ equals the semiperimeter of $ABCD$.

2007 Sharygin Geometry Olympiad, 15

Tags: geometry , equal angles , Angle Bisectors

In a triangle $ABC$, let $AA', BB'$ and $CC'$ be the bisectors. Suppose $A'B' \cap CC' =P$ and $A'C' \cap BB'= Q$. Prove that $\angle PAC = \angle QAB$.

2024 Israel TST, P1

Tags: geometry , Tangents , Angle Bisectors

Let $ABC$ be a triangle and let $D$ be a point on $BC$ so that $AD$ bisects the angle $\angle BAC$. The common tangents of the circles $(BAD)$, $(CAD)$ meet at the point $A'$. The points $B'$, $C'$ are defined similarly. Show that $A'$, $B'$, $C'$ are collinear.

2004 All-Russian Olympiad Regional Round, 11.8

Tags: geometry , 3D geometry , Angle Bisectors , angle bisector , pyramid , Spheres , sphere

Given a triangular pyramid $ABCD$. Sphere $S_1$ passing through points $A$, $B$, $C$, intersects edges $AD$, $BD$, $CD$ at points $K$, $L$, $M$, respectively; sphere $S_2$ passing through points $A$, $B$, $D$ intersects the edges $AC$, $BC$, $DC$ at points $P$, $Q$, $M$ respectively. It turned out that $KL \parallel PQ$. Prove that the bisectors of plane angles $KMQ$ and $LMP$ are the same.