Found problems: 81
Let $ABCDEFGHI$ be a regular polygon of $9$ sides. The segments $AE$ and $DF$ intersect at $P$. Prove that $PG$ and $AF$ are perpendicular.
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]
Gabriel plays to draw triangles using the vertices of a regular polygon with $2019$ sides, following these rules:
(i) The vertices used by each triangle must not have been previously used.
(ii) The sides of the triangle to be drawn must not intersect with the sides of the triangles previously drawn.
If Gabriel continues to draw triangles until it is no longer possible, determine the minimum number of triangles that he drew.
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 $\angle B_2A_1A_2$.
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?
If a polygon inscribed in a circle is equiangular and has an odd number of sides, prove that it is regular.
$A, B, C$ 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 $\angle XYC$.
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 $B, C$.
a) Prove that for $n = 2,3,4,...$ holds:
$$\sin a + \sin 2a + ...+ \sin (n-1)a=\frac{\cos a \left(\frac{a}{2}\right) - \cos \left(n-\frac{1}{2}\right) a}{2 \sin \left(\frac{a}{2}\right)}$$
b) A point on the circumference of a wheel, which, remaining in a vertical plane, rolls along a horizontal path, describes, at one revolution of the wheel, a curve having a length equal to four times the diameter of the wheel. Prove this by first considering tilting a regular $n$-gon.
[hide=original wording for part b]Een punt van de omtrek van een wiel dat, in een verticaal vlak blijvend, rolt over een horizontaal gedachte weg, beschrijft bij één omwenteling van het wiel een kromme die een lengte heeft die gelijk is aan viermaal de middellijn van het wiel.
Bewijs dit door eerst een rondkantelende regelmatige n-hoek te beschouwen.[/hide]
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.
Consider a regular $2n + 1$-gon $P$ 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 $SE$ 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 for which such a coloring is possible.
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 $QC$.
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?
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]
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.
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 $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$
A regular $1985$-gon is given in the plane. Let's pass a straight line through each side of it. Determine the number of parts into which these lines divide the plane.
Let $ABCDEFGHI$ be a regular polygon of $9$ sides. The segments $AE$ and $DF$ intersect at $P$. Prove that $PG$ and $AF$ are perpendicular.
(a) The vertices of a regular $10$-gon are painted in turn black and white. Two people play the following game . Each in turn draws a diagonal connecting two vertices of the same colour . These diagonals must not intersect . The winner is the player who is able to make the last move. Who will win if both players adopt the best strategy?
(b) Answer the same question for the regular $12$-gon .
(V.G. Ivanov)
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 $L$?
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?
$A, B, C, D$ are consecutive vertices of a regular $7$-gon. $AL$ and $AM$ are tangents to the circle center $C$ radius $CB$. $N$ is the intersection point of $AC$ and $BD$. Show that $L, M, N$ are collinear.
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.
(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 $O = N$.
A convex polygon $A_1A_2 . . .A_n$ of $n$ sides and inscribed in a circle, has its sides that satisfy the inequalities
$$A_nA_1 > A_1A_2 > A_2A_3 >...> A_{n-1}A_n$$
Show that its interior angles satisfy the inequalities $$\angle A_1 < \angle A_2 < \angle A_3 < ... < \angle A_{n-1}, \angle A_{n-1} > \angle A_n> \angle A_1.$$