Olympiad problems

2003 Estonia National Olympiad, 1

Let $A_1, A_2, ..., A_m$ and $B_2 , B_3,..., B_n$ be the points on a circle such that $A_1A_2... A_n$ is a regular $m$-gon and $A_1B_2...B_n$ is a regular $n$-gon whereby $n > m$ and the point $B_2$ lies between $A_1$ and $A_2$. Find $\angle B_2A_1A_2$.

2012 Czech-Polish-Slovak Junior Match, 4

Tags: polygon , regular polygon , combinatorics , combinatorial geometry , isosceles

Prove that among any $51$ vertices of the $101$-regular polygon there are three that are the vertices of an isosceles triangle.

2016 Sharygin Geometry Olympiad, P19

Tags: geometry , circumcircle , regular polygon , hexagon

Let $ABCDEF$ be a regular hexagon. Points $P$ and $Q$ on tangents to its circumcircle at $A$ and $D$ respectively are such that $PQ$ touches the minor arc $EF$ of this circle. Find the angle between $PB$ and $QC$.

2021 Israel TST, 1

Tags: geometry , regular polygon

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 $72^\circ$.

2004 Estonia Team Selection Test, 3

Tags: combinatorial geometry , combinatorics , regular polygon

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?

1983 Austrian-Polish Competition, 9

Tags: geometry , rectangle , symmetry , coloring , regular polygon , combinatorial geometry , combinatorics

To each side of the regular $p$-gon of side length $1$ there is attached a $1 \times k$ rectangle, partitioned into $k$ unit cells, where $k$ and $p$ are given positive integers and p an odd prime. Let $P$ be the resulting nonconvex star-like polygonal figure consisting of $kp + 1$ regions ($kp$ unit cells and the $p$-gon). Each region is to be colored in one of three colors, adjacent regions having different colors. Furthermore, it is required that the colored figure should not have a symmetry axis. In how many ways can this be done?

1998 All-Russian Olympiad Regional Round, 9.7

Tags: combinatorial geometry , geometry , regular polygon , combinatorics

Given a billiard in the form of a regular $1998$-gon $A_1A_2...A_{1998}$. A ball was released from the midpoint of side $A_1A_2$, which, reflected therefore from sides $A_2A_3$, $A_3A_4$, . . . , $A_{1998}A_1$ (according to the law, the angle of incidence is equal to the angle of reflection), returned to the starting point. Prove that the trajectory of the ball is a regular $1998$-gon.

2009 Estonia Team Selection Test, 3

Tags: polyhedron , regular polygon , convex , geometry , 3d geometry

Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions: (i) Each face is a regular polygon. (ii) Among the faces, there are polygons with at most two different numbers of edges. (iii) There are two faces with common edge that are both $n$-gons.

1982 Tournament Of Towns, (030) 4

Tags: geometry , 3d geometry , tetrahedron , vector , regular polygon , regular tetrahedron

(a) $K_1,K_2,..., K_n$ are the feet of the perpendiculars from an arbitrary point $M$ inside a given regular $n$-gon to its sides (or sides produced). Prove that the sum $\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n}$ equals $\frac{n}{2}\overrightarrow{MO}$, where $O$ is the centre of the $n$-gon. (b) Prove that the sum of the vectors whose origin is an arbitrary point $M$ inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from $M$ to the faces of the tetrahedron equals $\frac43 \overrightarrow{MO}$, where $O$ is the centre of the tetrahedron. (VV Prasolov, Moscow)

2001 Estonia Team Selection Test, 2

Tags: regular polygon , geometric inequalities , geometry , distance

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 $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$

2021 Final Mathematical Cup, 4

Tags: coloring , combinatorics , regular polygon

Let $P$ is a regular $(2n+1)$-gon in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ , if the line segment $\overline{ES}$ contains no other points that lie on the sides of $P$ except $S$ . We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$ , at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$ , we consider them colorless). Find the largest positive integer $n$ for which such a coloring is possible.

2012 Belarus Team Selection Test, 1

Tags: combinatorial geometry , geometry , combinatorics , regular polygon , regular

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)

2022 Iran Team Selection Test, 7

Tags: algebra , regular polygon

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

1967 Polish MO Finals, 5

Tags: geometry , polygon , regular polygon

Prove that if a cyclic polygon with an odd number of sides has all angles equal, then this polygon is regular.

2019 May Olympiad, 5

Tags: combinatorial geometry , combinatorics , regular polygon

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 Tournament of Towns, 6

Tags: geometry , combinatorial geometry , perpendicular , centroid , regular polygon

(a) A point $A$ is marked inside a circle. Two perpendicular lines drawn through $A$ intersect the circle at four points. Prove that the centre of mass of these four points does not depend on the choice of the lines. (b) A regular $2n$-gon ($n \ge 2$) with centre $A$ is drawn inside a circle (A does not necessarily coincide with the centre of the circle). The rays going from $A$ to the vertices of the $2n$-gon mark $2n$ points on the circle. Then the $2n$-gon is rotated about $A$. The rays going from $A$ to the new locations of vertices mark new $2n$ points on the circle. Let $O$ and $N$ be the centres of gravity of old and new points respectively. Prove that $O = N$.

1965 German National Olympiad, 4

Tags: inradius , locus , distance , geometry , regular polygon

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.

2006 Tournament of Towns, 1

Tags: heptagon , 17-gon , geometry , incircle , circumcircle , regular polygon , area

Two regular polygons, a $7$-gon and a $17$-gon are given. For each of them two circles are drawn, an inscribed circle and a circumscribed circle. It happened that rings containing the polygons have equal areas. Prove that sides of the polygons are equal. (3)

2022 Iran Team Selection Test, 7

Tags: algebra , regular polygon

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

2012 Tournament of Towns, 2

Tags: convex polyhedron , regular polygon , congruent , combinatorial geometry , geometry

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?

1983 Tournament Of Towns, (043) A5

Tags: coloring , combinatorics , combinatorial geometry , regular polygon

$k$ 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)

2021 Israel TST, 1

Tags: geometry , regular polygon

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 $72^\circ$.

1991 Chile National Olympiad, 2

Tags: inscribed , polygon , inscribed polygons , regular polygon , geometry

If a polygon inscribed in a circle is equiangular and has an odd number of sides, prove that it is regular.

2007 Sharygin Geometry Olympiad, 7

Tags: convex polygon , inscribed circle , regular polygon , geometry

A convex polygon is circumscribed around a circle. Points of contact of its sides with the circle form a polygon with the same set of angles (the order of angles may differ). Is it true that the polygon is regular?

2010 Sharygin Geometry Olympiad, 7

Tags: polygon , geometry , regular polygon , paper

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?