Olympiad problems

2023 Sharygin Geometry Olympiad, 13

The base $AD$ of a trapezoid $ABCD$ is twice greater than the base $BC$, and the angle $C$ equals one and a half of the angle $A$. The diagonal $AC$ divides angle $C$ into two angles. Which of them is greater?

2023 Sharygin Geometry Olympiad, 9.5

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023 , Isosceles Triangle

A point $D$ lie on the lateral side $BC$ of an isosceles triangle $ABC$. The ray $AD$ meets the line passing through $B$ and parallel to the base $AC$ at point $E$. Prove that the tangent to the circumcircle of triangle $ABD$ at $B$ bisects $EC$.

2023 Sharygin Geometry Olympiad, 9.1

Tags: Sharygin Geometry Olympiad , Sharygin 2023 , median , Concyclic , geometry

The ratio of the median $AM$ of a triangle $ABC$ to the side $BC$ equals $\sqrt{3}:2$. The points on the sides of $ABC$ dividing these side into $3$ equal parts are marked. Prove that some $4$ of these $6$ points are concyclic.

2023 Sharygin Geometry Olympiad, 5

Tags: geometry , perpendicular , Sharygin Geometry Olympiad , Sharygin 2023

Let $ABCD$ be a cyclic quadrilateral. Points $E$ and $F$ lie on the sides $AD$ and $CD$ in such a way that $AE = BC$ and $AB = CF$. Let $M$ be the midpoint of $EF$. Prove that $\angle AMC = 90^{\circ}$.

2023 Sharygin Geometry Olympiad, 10.3

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023

Let $\omega$ be the circumcircle of triangle $ABC$, $O$ be its center, $A'$ be the point of $\omega$ opposite to $A$, and $D$ be a point on a minor arc $BC$ of $\omega$. A point $D'$ is the reflection of $D$ about $BC$. The line $A'D'$ meets for the second time at point $E$. The perpendicular bisector to $D'E$ meets $AB$ and $AC$ at points $F$ and $G$ respectively. Prove that $\angle FOG = 180^\circ - 2\angle BAC$.

2023 Sharygin Geometry Olympiad, 18

Tags: geometry , bicentral , arc midpoint , constructive geometry , Sharygin Geometry Olympiad , Sharygin 2023

Restore a bicentral quadrilateral $ABCD$ if the midpoints of the arcs $AB,BC,CD$ of its circumcircle are given.

2023 Sharygin Geometry Olympiad, 4

Tags: geometry , incenter , Sharygin Geometry Olympiad , Sharygin 2023

Points $D$ and $E$ lie on the lateral sides $AB$ and $BC$ respectively of an isosceles triangle $ABC$ in such a way that $\angle BED = 3\angle BDE$. Let $D'$ be the reflection of $D$ about $AC$. Prove that the line $D'E$ passes through the incenter of $ABC$.

2023 Sharygin Geometry Olympiad, 1

Tags: geometry , Concyclic , Sharygin Geometry Olympiad , Sharygin 2023

Let $L$ be the midpoint of the minor arc $AC$ of the circumcircle of an acute-angled triangle $ABC$. A point $P$ is the projection of $B$ to the tangent at $L$ to the circumcircle. Prove that $P$, $L$, and the midpoints of sides $AB$, $BC$ are concyclic.

2023 Sharygin Geometry Olympiad, 21

Tags: geometry , humpty points , radical axis , projections , othorcenter , Sharygin Geometry Olympiad , Sharygin 2023

Let $ABCD$ be a cyclic quadrilateral; $M_{ac}$ be the midpoint of $AC$; $H_d,H_b$ be the orthocenters of $\triangle ABC,\triangle ADC$ respectively; $P_d,P_b$ be the projections of $H_d$ and $H_b$ to $BM_{ac}$ and $DM_{ac}$ respectively. Define similarly $P_a,P_c$ for the diagonal $BD$. Prove that $P_a,P_b,P_c,P_d$ are concyclic.

2023 Sharygin Geometry Olympiad, 8.7

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023

The bisector of angle $A$ of triangle $ABC$ meet its circumcircle $\omega$ at point $W$. The circle $s$ with diameter $AH$ ($H$ is the orthocenter of $ABC$) meets $\omega$ for the second time at point $P$. Restore the triangle $ABC$ if the points $A$, $P$, $W$ are given.

2023 Sharygin Geometry Olympiad, 8.6

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023

For which $n$ the plane may be paved by congruent figures bounded by $n$ arcs of circles?

2023 Sharygin Geometry Olympiad, 23

Tags: geometry , ellipse , de longchamps point , collinearity , Sharygin Geometry Olympiad , Sharygin 2023

An ellipse $\Gamma_1$ with foci at the midpoints of sides $AB$ and $AC$ of a triangle $ABC$ passes through $A$, and an ellipse $\Gamma_2$ with foci at the midpoints of $AC$ and $BC$ passes through $C$. Prove that the common points of these ellipses and the orthocenter of triangle $ABC$ are collinear.

2023 Sharygin Geometry Olympiad, 9.4

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023

The incircle $\omega$ of a triangle $ABC$ centered at $I$ touches $BC$ at point $D$. Let $P$ be the projection of the orthocenter of $ABC$ to the median from $A$. Prove that the circle $AIP$ and $\omega$ cut off equal chords on $AD$.

2023 Sharygin Geometry Olympiad, 10.8

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023

A triangle $ABC$ is given. Let $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ be circles centered at points $X$, $Y$, $Z$, $T$ respectively such that each of lines $BC$, $CA$, $AB$ cuts off on them four equal chords. Prove that the centroid of $ABC$ divides the segment joining $X$ and the radical center of $\omega_2$, $\omega_3$, $\omega_4$ in the ratio $2:1$ from $X$.

2023 Sharygin Geometry Olympiad, 7

Tags: geometry , Nine Point Circle , tangent circles , Sharygin Geometry Olympiad , Sharygin 2023

Let $A$ be a ﬁxed point of a circle $\omega$. Let $BC$ be an arbitrary chord of $\omega$ passing through a ﬁxed point $P$. Prove that the nine-points circles of triangles $ABC$ touch some ﬁxed circle not depending on $BC$.

2023 Sharygin Geometry Olympiad, 10.7

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023

There are $43$ points in the space: $3$ yellow and $40$ red. Any four of them are not coplanar. May the number of triangles with red vertices hooked with the triangle with yellow vertices be equal to $2023$? Yellow triangle is hooked with the red one if the boundary of the red triangle meet the part of the plane bounded by the yellow triangle at the unique point. The triangles obtained by the transpositions of vertices are identical.

2023 Sharygin Geometry Olympiad, 9.3

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023 , concurrent lines

Points $A_1$, $A_2$, $B_1$, $B_2$ lie on the circumcircle of a triangle $ABC$ in such a way that $A_1B_1 \parallel AB$, $A_1A_2 \parallel BC$, $B_1B_2 \parallel AC$. The line $AA_2$ and $CA_1$ meet at point $A'$, and the lines $BB_2$ and $CB_1$ meet at point $B'$. Prove that all lines $A'B'$ concur.

2023 Sharygin Geometry Olympiad, 8.1

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023

Let $ABC$ be an isosceles obtuse-angled triangle, and $D$ be a point on its base $AB$ such that $AD$ equals to the circumradius of triangle $BCD$. Find the value of $\angle ACD$.

2023 Sharygin Geometry Olympiad, 8.8

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023

Two circles $\omega_1$ and $\omega_2$ meeting at point $A$ and a line $a$ are given. Let $BC$ be an arbitrary chord of $\omega_2$ parallel to $a$, and $E$, $F$ be the second common points of $AB$ and $AC$ respectively with $\omega_1$. Find the locus of common points of lines $BC$ and $EF$.

2023 Sharygin Geometry Olympiad, 10.1

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023

Let $M$ be the midpoint of cathetus $AB$ of triangle $ABC$ with right angle $A$. Point $D$ lies on the median $AN$ of triangle $AMC$ in such a way that the angles $ACD$ and $BCM$ are equal. Prove that the angle $DBC$ is also equal to these angles.

2023 Sharygin Geometry Olympiad, 9.8

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023 , Angle Chasing

Let $ABC$ be a triangle with $\angle A = 120^\circ$, $I$ be the incenter, and $M$ be the midpoint of $BC$. The line passing through $M$ and parallel to $AI$ meets the circle with diameter $BC$ at points $E$ and $F$ ($A$ and $E$ lie on the same semiplane with respect to $BC$). The line passing through $E$ and perpendicular to $FI$ meets $AB$ and $AC$ at points $P$ and $Q$ respectively. Find the value of $\angle PIQ$.

2023 Sharygin Geometry Olympiad, 17

Tags: geometry , nagelian , concurrency , tangent circles , Sharygin Geometry Olympiad , Sharygin 2023

A common external tangent to circles $\omega_1$ and $\omega_2$ touches them at points $T_1, T_2$ respectively. Let $A$ be an arbitrary point on the extension of $T_1T_2$ beyond $T_1$, and $B$ be a point on the extension of $T_1T_2$ beyond $T_2$ such that $AT_1 = BT_2$. The tangents from $A$ to $\omega_1$ and from $B$ to $\omega_2$ distinct from $T_1T_2$ meet at point $C$. Prove that all nagelians of triangles $ABC$ from $C$ have a common point.

2023 Sharygin Geometry Olympiad, 10.4

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023

Let $ABC$ be a Poncelet triangle, $A_1$ is the reflection of $A$ about the incenter $I$, $A_2$ is isogonally conjugated to $A_1$ with respect to $ABC$. Find the locus of points $A_2$.

2023 Sharygin Geometry Olympiad, 16

Tags: geometry , reflection , altitudes , Sharygin Geometry Olympiad , Sharygin 2023

Let $AH_A$ and $BH_B$ be the altitudes of a triangle $ABC$. The line $H_AH_B$ meets the circumcircle of $ABC$ at points $P$ and $Q$. Let $A'$ be the reﬂection of $A$ about $BC$, and $B'$ be the reﬂection of $B$ about $CA$. Prove that $A',B', P,Q$ are concyclic.

2023 Sharygin Geometry Olympiad, 9.6

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023 , tangent circles , orthocenter

Let $ABC$ be acute-angled triangle with circumcircle $\Gamma$. Points $H$ and $M$ are the orthocenter and the midpoint of $BC$ respectively. The line $HM$ meets the circumcircle $\omega$ of triangle $BHC$ at point $N\not= H$. Point $P$ lies on the arc $BC$ of $\omega$ not containing $H$ in such a way that $\angle HMP = 90^\circ$. The segment $PM$ meets $\Gamma$ at point $Q$. Points $B'$ and $C'$ are the reflections of $A$ about $B$ and $C$ respectively. Prove that the circumcircles of triangles $AB'C'$ and $PQN$ are tangent.