Olympiad problems

2023 Sharygin Geometry Olympiad, 9.4

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 $AD$.

2020 Sharygin Geometry Olympiad, 21

Tags: Sharygin Geometry Olympiad , geometry

The diagonals of bicentric quadrilateral $ABCD$ meet at point $L$. Given are three segments equal to $AL$, $BL$, $CL$. Restore the quadrilateral using a compass and a ruler.

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 $BC$.

2010 Sharygin Geometry Olympiad, 6

Tags: geometry , angles , square , midpoints , Angle Chasing , geometry solved , Sharygin Geometry Olympiad

Let $E, F$ be the midpoints of sides $BC, CD$ of square $ABCD$. Lines $AE$ and $BF$ meet at point $P$. Prove that $\angle PDA = \angle AED$.

2022 Sharygin Geometry Olympiad, 7

Tags: Sharygin Geometry Olympiad , Sharygin 2022 , geometry

A square with center $F$ was constructed on the side $AC$ of triangle $ABC$ outside it. After this, everything was erased except $F$ and the midpoints $N,K$ of sides $BC,AB$. Restore the triangle.

2018 Oral Moscow Geometry Olympiad, 6

Tags: Sharygin Geometry Olympiad , geometry , circumcircle

Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circumcircle of $\triangle{BOC}$ meets the lines $AB, AC$ at points $A_1, A_2$, respectively. Let $\omega_{A}$ be the circumcircle of triangle $AA_1A_2$. Define $\omega_B$ and $\omega_C$ analogously. Prove that the circles $\omega_A, \omega_B, \omega_C$ concur on $\odot(ABC)$.

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.

2025 Sharygin Geometry Olympiad, 9

Tags: geometry , Sharygin Geometry Olympiad , Euler Line , ThalesTheorem

The line $l$ passing through the orthocenter $H$ of a triangle $ABC$ $(BC>AB)$ and parallel to $AC$ meets $AB$ and $BC$ at points $D$ and $E$ respectively. The line passing through the circumcenter of the triangle and parallel to the median $BM$ meets $l$ at point $F$. Prove that the length of segment $HF$ is three times greater than the difference of $FE$ and $DH$ Proposed by: A.Mardanov, K.Mardanova

2014 Sharygin Geometry Olympiad, 8

Tags: geometry , rectangle , Sharygin Geometry Olympiad

Let $ABCD$ be a rectangle. Two perpendicular lines pass through point $B$. One of them meets segment $AD$ at point $K$, and the second one meets the extension of side $CD$ at point $L$. Let $F$ be the common point of $KL$ and $AC$. Prove that $BF\perp KL$.

2022 Sharygin Geometry Olympiad, 10

Tags: Sharygin Geometry Olympiad , Sharygin 2022 , geometry

Let $\omega_1$ be the circumcircle of triangle $ABC$ and $O$ be its circumcenter. A circle $\omega_2$ touches the sides $AB, AC$, and touches the arc $BC$ of $\omega_1$ at point $K$. Let $I$ be the incenter of $ABC$. Prove that the line $OI$ contains the symmedian of triangle $AIK$.

2022 Sharygin Geometry Olympiad, 6

Tags: Sharygin Geometry Olympiad , Sharygin 2022 , geometric inequality , inequalities

The incircle and the excircle of triangle $ABC$ touch the side $AC$ at points $P$ and $Q$ respectively. The lines $BP$ and $BQ$ meet the circumcircle of triangle $ABC$ for the second time at points $P'$ and $Q'$ respectively. Prove that $$PP' > QQ'$$

2025 Sharygin Geometry Olympiad, 5

Tags: geometry , Sharygin Geometry Olympiad , tangent

Let $M$ be the midpoint of the cathetus $AC$ of a right-angled triangle $ABC$ $(\angle C=90^{\circ})$. The perpendicular from $M$ to the bisector of angle $ABC$ meets $AB$ at point $N$. Prove that the circumcircle of triangle $ANM$ touches the bisector of angle $ABC$. Proposed by:D.Shvetsov

2024 Sharygin Geometry Olympiad, 8.3

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2024

Let $AD$ be the altitude of an acute-angled triangle $ABC$ and $A'$ be the point on its circumcircle opposite to $A$. A point $P$ lies on the segment $AD$, and points $X$, $Y$ lie on the segments $AB$, $AC$ respectively in such a way that $\angle CBP = \angle ADY$, $\angle BCP = \angle ADX$. Let $PA'$ meet $BC$ at point $T$. Prove that $D$, $X$, $Y$, $T$ are concyclic.

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.

2019 Sharygin Geometry Olympiad, 6

Tags: geometry , Sharygin Geometry Olympiad

A point $H$ lies on the side $AB$ of regular polygon $ABCDE$. A circle with center $H$ and radius $HE$ meets the segments $DE$ and $CD$ at points $G$ and $F$ respectively. It is known that $DG=AH$. Prove that $CF=AH$.

2024 Sharygin Geometry Olympiad, 8.5

Tags: geometry , rectangle , Sharygin Geometry Olympiad , Sharygin 2024

The vertices $M$, $N$, $K$ of rectangle $KLMN$ lie on the sides $AB$, $BC$, $CA$ respectively of a regular triangle $ABC$ in such a way that $AM = 2$, $KC = 1$. The vertex $L$ lies outside the triangle. Find the value of $\angle KMN$.

2024 Sharygin Geometry Olympiad, 8.8

Tags: combinatorial geometry , Sharygin Geometry Olympiad , Sharygin 2024 , geometry

Two polygons are cut from the cardboard. Is it possible that for any disposition of these polygons on the plane they have either common inner points or only a finite number of common points on the boundary?

2019 Sharygin Geometry Olympiad, 1

Tags: geometry , Sharygin Geometry Olympiad , circumcircle

A trapezoid with bases $AB$ and $CD$ is inscribed into a circle centered at $O$. Let $AP$ and $AQ$ be the tangents from $A$ to the circumcircle of triangle $CDO$. Prove that the circumcircle of triangle $APQ$ passes through the midpoint of $AB$.

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 Myanmar IMO Training, 6

Tags: geometry , Sharygin Geometry Olympiad

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.

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.

2022 Sharygin Geometry Olympiad, 15

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2022 , Inversion , similar triangles

A line $l$ parallel to the side $BC$ of triangle $ABC$ touches its incircle and meets its circumcircle at points $D$ and $E$. Let $I$ be the incenter of $ABC$. Prove that $AI^2 = AD \cdot AE$.

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 $\angle PIQ$.