Let there be a regular polygon of $n$ sides with center $O$. Determine the highest possible number of vertices $k$ $(k \geq 3)$, which can be coloured in green, such that $O$ is strictly outside of any triangle with $3$ vertices coloured green. Determine this $k$ for $a) n=2019$ ; $b) n=2020$.

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 $A,R$

In a circle with radius 1 a regular n-gon $A_1 A_2...A_n$ is inscribed. Calculate the product: $A_1 A_2.A_1 A_3 \dots A_1 A_{n-1} .A_1 A_n$.

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.

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

In the complex plane consider the unit circle with the origin as its center. Furthermore, consider inscribed regular 17-gon with one of its vertices being $1+0i.$ How many of its vertices lie in the (open) unit disc centered in $\sqrt{3/2}(1+i)$?

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) $N = 2012$ ?

For which integers $n \ge 5$ is it possible to color the vertices of a regular$ n$-gon using at most $6$ colors in such a way that any $5$ consecutive vertices have different colors?

Let $P$ a convex regular polygon with $n$ sides, having the center $O$ and $\angle xOy$ an angle of measure $a$, $a \in (0,k)$. Let $S$ be the area of the common part of the interiors of the polygon and the angle. Find, as a function of $n$, the values of $a$ such that $S$ remains constant when $\angle xOy$ is rotating around $O$.

Given is a regular $n$-gon with circumradius $1$. $L$ is the set of (different) lengths of all connecting segments of its endpoints. What is the sum of the squares of the elements of $L$?

(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)

A regular $2017$-gon is partitioned into triangles by a set of non-intersecting diagonals. Prove that among those triangles only one is acute-angled.

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.

Let $P$ be a regular polygon with $ 4k + 1 $ sides (where $ k $ is a natural) whose vertices are $ A_1, A_2, ..., A_ {4k + 1} $ (in that order ). Each vertex $ A_j $ of $P$ is assigned a natural of the set $ \{1,2, ..., 4k + 1 \} $ such that no two vertices are assigned the same number. On $P$ the following operation is performed: Let $ B_j $ be the midpoint of the side $ A_jA_ {j + 1} $ for $ j = 1,2, ..., 4k + 1 $ (where is consider $ A_ {4k + 2} = A_1 $). If $ a $, $ b $ are the numbers assigned to $ A_ {j} $ and $ A_ {j + 1} $, respectively, the midpoint $ B_j $ is written the number $ 7a-3b $. By doing this with each of the $ 4k + 1 $ sides, the $ 4k + 1 $ vertices initially arranged are erased. We will say that a natural $ m $ is [i]fatal [/i] if for all natural $ k $ , no matter how the vertices of $P$ are initially arranged, it is impossible to obtain $ 4k + 1 $ equal numbers through a finite amount of operations from $ m $. a) Determine if the $ 2010 $ is fatal or not. Justify. b) Prove that there are infinite fatal numbers. [color=#f00]PS. A help in translation of the 2nd paragraph is welcome[/color]. [hide=Original wording]Diremos que un natural $m$ es fatal si no importa cómo se disponen inicialmente los vértices de ${P}$, es imposible obtener mediante una cantidad finita de operaciones $4k+1$ números iguales a $m$.[/hide]

$A, B, C$ are adjacent vertices of a regular $2n$-gon and $D$ is the vertex opposite to $B$ (so that $BD$ passes through the center of the $2n$-gon). $X$ is a point on the side $AB$ and $Y$ is a point on the side $BC$ so that $XDY = \frac{\pi}{2n}$. Show that $DY$ bisects $\angle XYC$.

Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.

(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)

Every two of five regular pentagons on the plane have a common point. Is it true that some of these pentagons have a common point?

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)

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?

$k$ 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)

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 $72^\circ$.

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]

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.

Is it possible to dissect a regular decagon along some of its diagonals so that the resulting parts can form two regular polygons? by N.Beluhov