Olympiad problems

2019 Sharygin Geometry Olympiad, 7

Let points $M$ and $N$ lie on sides $AB$ and $BC$ of triangle $ABC$ in such a way that $MN||AC$. Points $M'$ and $N'$ are the reflections of $M$ and $N$ about $BC$ and $AB$ respectively. Let $M'A$ meet $BC$ at $X$, and let $N'C$ meet $AB$ at $Y$. Prove that $A,C,X,Y$ are concyclic.

2017 Sharygin Geometry Olympiad, 5

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

Let $BH_b, CH_c$ be altitudes of an acute-angled triangle $ABC$. The line $H_bH_c$ meets the circumcircle of $ABC$ at points $X$ and $Y$. Points $P,Q$ are the reflections of $X,Y$ about $AB,AC$ respectively. Prove that $PQ \parallel BC$. [i]Proposed by Pavel Kozhevnikov[/i]

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.

2024 Sharygin Geometry Olympiad, 8.4

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2024

A square with side $1$ is cut from the paper. Construct a segment with length $1/2024$ using at most $20$ folds. No instruments are available. It is allowed only to fold the paper and to mark the common points of folding lines.

2013 Sharygin Geometry Olympiad, 20

Tags: analytic geometry , geometry , Sharygin Geometry Olympiad

Let $C_1$ be an arbitrary point on the side $AB$ of triangle $ABC$. Points $A_1$ and $B_1$ on the rays $BC$ and $AC$ are such that $\angle AC_1B_1 = \angle BC_1A_1 = \angle ACB$. The lines $AA_1$ and $BB_1$ meet in point $C_2$. Prove that all the lines $C_1C_2$ 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 $A_2$.

2019 Sharygin Geometry Olympiad, 2

Tags: geometry , Sharygin Geometry Olympiad

A point $M$ inside triangle $ABC$ is such that $AM=AB/2$ and $CM=BC/2$. Points $C_0$ and $A_0$ lying on $AB$ and $CB$ respectively are such that $BC_0:AC_0 = BA_0:CA_0 = 3$. Prove that the distances from $M$ to $C_0$ and $A_0$ are equal.

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.

2023 Sharygin Geometry Olympiad, 12

Tags: geometry , perpendicular bisector , orthologic triangles , Sharygin Geometry Olympiad , Sharygin 2023

Let $ABC$ be a triangle with obtuse angle $B$, and $P, Q$ lie on $AC$ in such a way that $AP = PB, BQ = QC$. The circle $BPQ$ meets the sides $AB$ and $BC$ at points $N$ and $M$ respectively. $\qquad\textbf{(a)}$ (grades 8-9) Prove that the distances from the common point $R$ of $PM$ and $NQ$ to $A$ and $C$ are equal. $\qquad\textbf{(b)}$ (grades 10-11) Let $BR$ meet $AC$ at point $S$. Prove that $MN \perp OS$, where $O$ is the circumcenter of $ABC$.

2023 Sharygin Geometry Olympiad, 10.2

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

The Euler line of a scalene triangle touches its incircle. Prove that this triangle is obtuse-angled.

2023 Sharygin Geometry Olympiad, 9.7

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

Let $H$ be the orthocenter of triangle $\mathrm T$. The sidelines of triangle $\mathrm T_1$ pass through the midpoints of $\mathrm T$ and are perpendicular to the corresponding bisectors of $\mathrm T$. The vertices of triangle $\mathrm T_2$ bisect the bisectors of $\mathrm T$. Prove that the lines joining $H$ with the vertices of $\mathrm T_1$ are perpendicular to the sidelines of $\mathrm T_2$.

2023 Sharygin Geometry Olympiad, 6

Tags: geometry , mittenpunkt , concurrency , Sharygin Geometry Olympiad , Sharygin 2023

Let $A_1, B_1, C_1$ be the feet of altitudes of an acute-angled triangle $ABC$. The incircle of triangle $A_1B_1C_1$ touches $A_1B_1, A_1C_1, B_1C_1$ at points $C_2, B_2, A_2$ respectively. Prove that the lines $AA_2, BB_2, CC_2$ concur at a point lying on the Euler line of triangle $ABC$.

2023 Sharygin Geometry Olympiad, 11

Tags: geometry , othorcenter , antipode , Sharygin Geometry Olympiad , Sharygin 2023

Let $H$ be the orthocenter of an acute-angled triangle $ABC$; $E$, $F$ be points on $AB, AC$ respectively, such that $AEHF$ is a parallelogram; $X, Y$ be the common points of the line $EF$ and the circumcircle $\omega$ of triangle $ABC$; $Z$ be the point of $\omega$ opposite to $A$. Prove that $H$ is the orthocenter of triangle $XYZ$.

2023 Sharygin Geometry Olympiad, 3

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

A circle touches the lateral sides of a trapezoid $ABCD$ at points $B$ and $C$, and its center lies on $AD$. Prove that the diameter of the circle is less than the medial line of the trapezoid.

2013 Sharygin Geometry Olympiad, 21

Tags: geometry , Sharygin Geometry Olympiad , geometry solved , power of a point , reflection , Angle Chasing , similar triangles

Chords $BC$ and $DE$ of circle $\omega$ meet at point $A$. The line through $D$ parallel to $BC$ meets $\omega$ again at $F$, and $FA$ meets $\omega$ again at $T$. Let $M = ET \cap BC$ and let $N$ be the reflection of $A$ over $M$. Show that $(DEN)$ passes through the midpoint of $BC$.

2023 Sharygin Geometry Olympiad, 19

Tags: geometry , Locus , circle intersections , Sharygin Geometry Olympiad , Sharygin 2023

A cyclic quadrilateral $ABCD$ is given. An arbitrary circle passing through $C$ and $D$ meets $AC,BC$ at points $X,Y$ respectively. Find the locus of common points of circles $CAY$ and $CBX$.

2023 Sharygin Geometry Olympiad, 8.2

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023 , similar triangles

The bisectors of angles $A$, $B$, and $C$ of triangle $ABC$ meet for the second time its circumcircle at points $A_1$, $B_1$, $C_1$ respectively. Let $A_2$, $B_2$, $C_2$ be the midpoints of segments $AA_1$, $BB_1$, $CC_1$ respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.

2023 Sharygin Geometry Olympiad, 8.5

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023

The median $CM$ and the altitude $AH$ of an acute-angled triangle $ABC$ meet at point $O$. A point $D$ lies outside the triangle in such a way that $AOCD$ is a parallelogram. Find the length of $BD$, if $MO= a$, $OC = b$.

2023 Sharygin Geometry Olympiad, 8

Tags: geometry , construction , length , Sharygin Geometry Olympiad , Sharygin 2023

A triangle $ABC$ $(a>b>c)$ is given. Its incenter $I$ and the touching points $K, N$ of the incircle with $BC$ and $AC$ respectively are marked. Construct a segment with length $a-c$ using only a ruler and drawing at most three lines.

2024 Sharygin Geometry Olympiad, 8.2

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2024

Let $CM$ be the median of an acute-angled triangle $ABC$, and $P$ be the projection of the orthocenter $H$ to the bisector of $\angle C$. Prove that $MP$ bisects the segment $CH$.

2023 Sharygin Geometry Olympiad, 10

Tags: geometry , angle bisector , tangent circles , Sharygin Geometry Olympiad , Sharygin 2023

Altitudes $BE$ and $CF$ of an acute-angled triangle $ABC$ meet at point $H$. The perpendicular from $H$ to $EF$ meets the line $\ell$ passing through $A$ and parallel to $BC$ at point $P$. The bisectors of two angles between $\ell$ and $HP$ meet $BC$ at points $S$ and $T$. Prove that the circumcircles of triangles $ABC$ and $PST$ are tangent.

2016 Sharygin Geometry Olympiad, 1

Tags: geometry , circumcircle , Sharygin Geometry Olympiad

A line parallel to the side $BC$ of a triangle $ABC$ meets the sides $AB$ and $AC$ at points $P$ and $Q$, respectively. A point $M$ is chosen inside the triangle $APQ$. The segments $MB$ and $MC$ meet the segment $PQ$ at points $E$ and $F$, respectively. Let $N$ be the second intersection point of the circumcircles of the triangles $PMF$ and $QME$. Prove that the points $A,M,N$ are collinear.

2023 Sharygin Geometry Olympiad, 9.2

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023 , combinatorial geometry , hexagon

Can a regular triangle be placed inside a regular hexagon in such a way that all vertices of the triangle were seen from each vertex of the hexagon? (Point $A$ is seen from $B$, if the segment $AB$ dots not contain internal points of the triangle.)

2023 Sharygin Geometry Olympiad, 10.5

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023

The incircle of a triangle $ABC$ touches $BC$ at point $D$. Let $M$ be the midpoint of arc $\widehat{BAC}$ of the circumcircle, and $P$, $Q$ be the projections of $M$ to the external bisectors of angles $B$ and $C$ respectively. Prove that the line $PQ$ bisects $AD$.