This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 8

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.

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 $X'Y'$.

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.

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?

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.

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 $CH$.