Found problems: 81
2001 Estonia Team Selection Test, 2
Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$
2004 Estonia Team Selection Test, 3
For which natural number $n$ is it possible to draw $n$ line segments between vertices of a regular $2n$-gon so that every vertex is an endpoint for exactly one segment and these segments have pairwise different lengths?
1990 Chile National Olympiad, 6
Given a regular polygon with apothem $ A $ and circumradius $ R $. Find for a regular polygon of equal perimeter and with double number of sides, the apothem $ a $ and the circumcircle $ r $ in terms of $A,R$
1967 Polish MO Finals, 5
Prove that if a cyclic polygon with an odd number of sides has all angles equal, then this polygon is regular.
1983 Tournament Of Towns, (033) O2
(a) A regular $4k$-gon is cut into parallelograms. Prove that among these there are at least $k$ rectangles.
(b) Find the total area of the rectangles in (a) if the lengths of the sides of the $4k$-gon equal $a$.
(VV Proizvolov, Moscow)
2010 Sharygin Geometry Olympiad, 7
Each of two regular polygons $P$ and $Q$ was divided by a line into two parts. One part of $P$ was attached to one part of $Q$ along the dividing line so that the resulting polygon was regular and not congruent to $P$ or $Q$. How many sides can it have?
2023 Bangladesh Mathematical Olympiad, P9
Let $A_1A_2\dots A_{2n}$ be a regular $2n$-gon inscribed in circle $\omega$. Let $P$ be any point on the circle $\omega$. Let $H_1,H_2,\dots, H_n$ be orthocenters of triangles $PA_1A_2, PA_3A_4,\dots, PA_{2n-1}A_{2n}$ respectively. Prove that $H_1H_2\dots H_n$ is a regular $n$-gon.
May Olympiad L2 - geometry, 2019.5
We consider the $n$ vertices of a regular polygon with $n$ sides. There is a set of triangles with vertices at these $n$ points with the property that for each triangle in the set, the sides of at least one are not the side of any other triangle in the set. What is the largest amount of triangles that can have the set?
[hide=original wording]Consideramos los n vértices de un polígono regular de n lados. Se tiene un conjunto de triángulos con vértices en estos n puntos con la propiedad que para cada triángulo del conjunto, al menos uno
de sus lados no es lado de ningún otro triángulo del conjunto. ¿Cuál es la mayor cantidad de triángulos que puede tener el conjunto?[/hide]
2012 Belarus Team Selection Test, 1
Let $m,n,k$ be pairwise relatively prime positive integers greater than $3$.
Find the minimal possible number of points on the plane with the following property:
there are $x$ of them which are the vertices of a regular $x$-gon for $x = m, x = n, x = k$.
(E.Piryutko)
2012 Tournament of Towns, 2
Given a convex polyhedron and a sphere intersecting each its edge at two points so that each edge is trisected (divided into three equal parts). Is it necessarily true that all faces of the polyhedron are
(a) congruent polygons?
(b) regular polygons?
2019 May Olympiad, 5
We consider the $n$ vertices of a regular polygon with $n$ sides. There is a set of triangles with vertices at these $n$ points with the property that for each triangle in the set, the sides of at least one are not the side of any other triangle in the set. What is the largest amount of triangles that can have the set?
[hide=original wording]Consideramos los n vértices de un polígono regular de n lados. Se tiene un conjunto de triángulos con vértices en estos n puntos con la propiedad que para cada triángulo del conjunto, al menos uno
de sus lados no es lado de ningún otro triángulo del conjunto. ¿Cuál es la mayor cantidad de triángulos que puede tener el conjunto?[/hide]
1965 German National Olympiad, 4
Find the locus of points in the plane, the sum of whose distances from the sides of a regular polygon is five times the inradius of the pentagon.
2016 Israel National Olympiad, 6
Points $A_1,A_2,A_3,...,A_{12}$ are the vertices of a regular polygon in that order. The 12 diagonals $A_1A_6,A_2A_7,A_3A_8,...,A_{11}A_4,A_{12}A_5$ are marked, as in the picture below. Let $X$ be some point in the plane. From $X$, we draw perpendicular lines to all 12 marked diagonals. Let $B_1,B_2,B_3,...,B_{12}$ be the feet of the perpendiculars, so that $B_1$ lies on $A_1A_6$, $B_2$ lies on $A_2A_7$ and so on.
Evaluate the ratio $\frac{XA_1+XA_2+\dots+XA_{12}}{B_1B_6+B_2B_7+\dots+B_{12}B_5}$.
[img]https://i.imgur.com/DUuwFth.png[/img]
1997 Czech And Slovak Olympiad IIIA, 2
Each side and diagonal of a regular $n$-gon ($n \ge 3$) for odd $n$ is colored red or blue. One may choose a vertex and change the color of all segments emanating from that vertex. Prove that, no matter how the edges were colored initially, one can achieve that the number of blue segments at each vertex is even. Prove also that the resulting coloring depends only on the initial coloring.
2021 Israel TST, 1
Let $ABCDEFGHIJ$ be a regular $10$-gon. Let $T$ be a point inside the $10$-gon, such that the $DTE$ is isosceles: $DT = ET$ , and its angle at the apex is $72^\circ$. Prove that there exists a point $S$ such that $FTS$ and $HIS$ are both isosceles, and for both of them the angle at the apex is $72^\circ$.
2013 Korea Junior Math Olympiad, 8
Drawing all diagonals in a regular $2013$-gon, the regular $2013$-gon is divided into non-overlapping polygons. Prove that there exist exactly one $2013$-gon out of all such polygons.
2012 Sharygin Geometry Olympiad, 4
Determine all integer $n > 3$ for which a regular $n$-gon can be divided into equal triangles by several (possibly intersecting) diagonals.
(B.Frenkin)
2004 Estonia Team Selection Test, 3
For which natural number $n$ is it possible to draw $n$ line segments between vertices of a regular $2n$-gon so that every vertex is an endpoint for exactly one segment and these segments have pairwise different lengths?
2011 Tournament of Towns, 4
Does there exist a convex $N$-gon such that all its sides are equal and all vertices belong to the parabola $y = x^2$ for
a) $N = 2011$
b) $N = 2012$ ?
2015 Estonia Team Selection Test, 8
Find all positive integers $n$ for which it is possible to partition a regular $n$-gon into triangles with diagonals not intersecting inside the $n$-gon such that at every vertex of the $n$-gon an odd number of triangles meet.
2021 Saudi Arabia Training Tests, 28
Find all positive integer $n\ge 3$ such that it is possible to mark the vertices of a regular $n$- gon with the number from 1 to n so that for any three vertices $A, B$ and $C$ with $AB = AC$, the number in $A$ is greater or smaller than both numbers in $B, C$.
1983 Tournament Of Towns, (043) A5
$k$ vertices of a regular $n$-gon $P$ are coloured. A colouring is called almost uniform if for every positive integer $m$ the following condition is satisfied:
If $M_1$ is a set of m consecutive vertices of $P$ and $M_2$ is another such set then the number of coloured vertices of $M_1$ differs from the number of coloured vertices of $M_2$ at most by $1$.
Prove that for all positive integers $k$ and $n$ ($k \le n$) an almost uniform colouring exists and that it is unique within a rotation.
(M Kontsevich, Moscow)
1974 Spain Mathematical Olympiad, 8
The sides of a convex regular polygon of $L + M + N$ sides are to be given draw in three colors: $L$ of them with a red stroke, $M$ with a yellow stroke, and $N$ with a blue. Express, through inequalities, the necessary and sufficient conditions so that there is a solution (several, in general) to the problem of doing it without leaving two adjacent sides drawn with the same color.
1983 Tournament Of Towns, (042) O5
A point is chosen inside a regular $k$-gon in such a way that its orthogonal projections on to the sides all meet the respective sides at interior points. These points divide the sides into $2k$ segments. Let these segments be enumerated consecutively by the numbers $1,2, 3, ... ,2k$. Prove that the sum of the lengths of the segments having even numbers equals the sum of the segments having odd numbers.
(A Andjans, Riga)
2022 Iran Team Selection Test, 7
Suppose that $n$ is a positive integer number. Consider a regular polygon with $2n$ sides such that one of its largest diagonals is parallel to the $x$-axis. Find the smallest integer $d$ such that there is a polynomial $P$ of degree $d$ whose graph intersects all sides of the polygon on points other than vertices.
Proposed by Mohammad Ahmadi