Olympiad problems

2025 Sharygin Geometry Olympiad, 15

A point $C$ lies on the bisector of an acute angle with vertex $S$. Let $P$, $Q$ be the projections of $C$ to the sidelines of the angle. The circle centered at $C$ with radius $PQ$ meets the sidelines at points $A$ and $B$ such that $SA\ne SB$. Prove that the circle with center $A$ touching $SB$ and the circle with center $B$ touching $SA$ are tangent. Proposed by: A.Zaslavsky

2024 Sharygin Geometry Olympiad, 8.1

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2024

A circle $\omega$ centered at $O$ and a point $P$ inside it are given. Let $X$ be an arbitrary point of $\omega$, the line $XP$ and the circle $XOP$ meet $\omega$ for a second time at points $X_1$, $X_2$ respectively. Prove that all lines $X_1X_2$ are parallel.

2011 Sharygin Geometry Olympiad, 18

Tags: geometry , combinatorial geometry , lines , Dissecting plane , Sharygin Geometry Olympiad

On the plane, given are $n$ lines in general position, i.e. any two of them aren’t parallel and any three of them don’t concur. These lines divide the plane into several parts. What is a) the minimal, b) the maximal number of these parts that can be angles?

2021 Sharygin Geometry Olympiad, 13

Tags: geometry , excircle , incenter , mixtilinear , concurrency , Sharygin Geometry Olympiad , anant mudgal geo

In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.

2025 Sharygin Geometry Olympiad, 1

Tags: geometry , Sharygin Geometry Olympiad , Angle Chasing

Let $I$ be the incenter of a triangle $ABC$, $D$ be an arbitrary point of segment $AC$, and $A_{1}, A_{2}$ be the common points of the perpendicular from $D$ to the bisector $CI$ with $BC$ and $AI$ respectively. Define similarly the points $C_{1}$, $C_{2}$. Prove that $B$, $A_{1}$, $A_{2}$, $I$, $C_{1},$ $C_{2}$ are concyclic. Proposed by:D.Shvetsov

2019 Sharygin Geometry Olympiad, 1

Tags: geometry , Sharygin Geometry Olympiad

A triangle $OAB$ with $\angle A=90^{\circ}$ lies inside another triangle with vertex $O$. The altitude of $OAB$ from $A$ until it meets the side of angle $O$ at $M$. The distances from $M$ and $B$ to the second side of angle $O$ are $2$ and $1$ respectively. Find the length of $OA$.

2019 Sharygin Geometry Olympiad, 8

Tags: geometry , Sharygin Geometry Olympiad

What is the least positive integer $k$ such that, in every convex 1001-gon, the sum of any k diagonals is greater than or equal to the sum of the remaining diagonals?

2022 Sharygin Geometry Olympiad, 8

Tags: Sharygin Geometry Olympiad , Sharygin 2022 , geometry

Points $P,Q,R$ lie on the sides $AB,BC,CA$ of triangle $ABC$ in such a way that $AP=PR, CQ=QR$. Let $H$ be the orthocenter of triangle $PQR$, and $O$ be the circumcenter of triangle $ABC$. Prove that $$OH||AC$$.

2015 Sharygin Geometry Olympiad, 6

Tags: geometry , perpendicular bisector , perpendicular , geometry solved , Sharygin Geometry Olympiad , Russia , median

Lines $b$ and $c$ passing through vertices $B$ and $C$ of triangle $ABC$ are perpendicular to sideline $BC$. The perpendicular bisectors to $AC$ and $AB$ meet $b$ and $c$ at points $P$ and $Q$ respectively. Prove that line $PQ$ is perpendicular to median $AM$ of triangle $ABC$. (D. Prokopenko)

2023 Sharygin Geometry Olympiad, 13

Tags: geometry , trapezoid , angles , Sharygin Geometry Olympiad , Sharygin 2023

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 $EC$.

2024 Sharygin Geometry Olympiad, 8.7

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2024

A convex quadrilateral $ABCD$ is given. A line $l \parallel AC$ meets the lines $AD$, $BC$, $AB$, $CD$ at points $X$, $Y$, $Z$, $T$ respectively. The circumcircles of triangles $XYB$ and $ZTB$ meet for the second time at point $R$. Prove that $R$ lies on $BD$.

2024 Sharygin Geometry Olympiad, 8.6

Tags: geometry , circumcircle , Sharygin Geometry Olympiad , Sharygin 2024 , reflection

A circle $\omega$ touched lines $a$ and $b$ at points $A$ and $B$ respectively. An arbitrary tangent to the circle meets $a$ and $b$ at $X$ and $Y$ respectively. Points $X'$ and $Y'$ are the reflections of $X$ and $Y$ about $A$ and $B$ respectively. Find the locus of projections of the center of the circle to the lines $X'Y'$.

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.

2019 Sharygin Geometry Olympiad, 3

Tags: geometry , Sharygin Geometry Olympiad , moving points

Let $ABCD$ be a cyclic quadrilateral such that $AD=BD=AC$. A point $P$ moves along the circumcircle $\omega$ of triangle $ABCD$. The lined $AP$ and $DP$ meet the lines $CD$ and $AB$ at points $E$ and $F$ respectively. The lines $BE$ and $CF$ meet point $Q$. Find the locus of $Q$.

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?

2012 Sharygin Geometry Olympiad, 22

Tags: geometry , circumcircle , cyclic quadrilateral , angle bisector , Sharygin Geometry Olympiad

A circle $\omega$ with center $I$ is inscribed into a segment of the disk, formed by an arc and a chord $AB$. Point $M$ is the midpoint of this arc $AB$, and point $N$ is the midpoint of the complementary arc. The tangents from $N$ touch $\omega$ in points $C$ and $D$. The opposite sidelines $AC$ and $BD$ of quadrilateral $ABCD$ meet in point $X$, and the diagonals of $ABCD$ meet in point $Y$. Prove that points $X, Y, I$ and $M$ are collinear.

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.