Found problems: 200
Let $A_1 A_2...A_n$ be a convex $n$-gon. What’s the number of $m$-gons with vertices from $A_1,A_2,...,A_n$ such that between each two adjacent vertices of the $m$-gon there are at least $k$ vertices from the $n$-gon?
Consider $7$-gons inscribed in a circle such that all sides of the $7$-gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$-gon.
Show that any two points lying inside a regular $ n\minus{}$gon $ E$ can be joined by two circular arcs lying inside $ E$ and meeting at an angle of at least $ \left(1 \minus{} \frac{2}{n} \right) \cdot \pi.$
Let $P$ be the set of all polygons in the plane and let $M : P \to R$ be a mapping that satisfies:
(i) $M(P) \ge 0$ for each polygon $P$,
(ii) $M(P) = x^2$ if $P$ is an equilateral triangle of side $x$,
(iii) If a polygon $P$ is partitioned into polygons $S$ and $T$, then $M(P) = M(S)+ M(T)$,
(iv) If polygons $P$ and $T$ are congruent, then $M(P) = M(T )$.
Determine $M(P)$ if $P$ is a rectangle with edges $x$ and $y$.
A [i]triangulation[/i] of a convex polygon $\Pi$ is a partitioning of $\Pi$ into triangles by diagonals having no common points other than the vertices of the polygon. We say that a triangulation is a [i]Thaiangulation[/i] if all triangles in it have the same area.
Prove that any two different Thaiangulations of a convex polygon $\Pi$ differ by exactly two triangles. (In other words, prove that it is possible to replace one pair of triangles in the first Thaiangulation with a different pair of triangles so as to obtain the second Thaiangulation.)
[i]Proposed by Bulgaria[/i]
A circle of length $10$ is inscribed in a convex polygon with perimeter $15$. What part of the area of this polygon is occupied by the resulting circle?
Each of Mahdi and Morteza has drawn an inscribed $93$-gon. Denote the first one by $A_1A_2…A_{93}$ and the second by $B_1B_2…B_{93}$. It is known that $A_iA_{i+1} // B_iB_{i+1}$ for $1 \le i \le 93$ ($A_{93} = A_1, B_{93} = B_1$). Show that $\frac{A_iA_{i+1} }{ B_iB_{i+1}}$ is a constant number independent of $i$.
by Morteza Saghafian
Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\]
where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively.
[i]Proposed by Witold Szczechla, Poland[/i]
It is given regular polygon with $2n$ sides and center $S$. Consider every quadrilateral with vertices as vertices of polygon. Let $u$ be number of such quadrilaterals which contain point $S$ inside and $v$ number of remaining quadrilaterals. Find $u-v$
A regular $17$-gon with vertices $V_1, V_2, . . . , V_{17}$ and sides of length $3$ has a point $ P$ on $V_1V_2$ such that $V_1P = 1$. A chord that stretches from $V_1$ to $V_2$ containing $ P$ is rotated within the interior of the heptadecagon around $V_2$ such that the chord now stretches from $V_2$ to $V_3$. The chord then hinges around $V_3$, then $V_4$, and so on, continuing until $ P$ is back at its original position. Find the total length traced by $ P$.
Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles.
[i]Proposed by Juozas Juvencijus Macys, Lithuania[/i]
All angles of the convex $n$-gon $A_1A_2\dots A_n$ are obtuse, where $n\ge5$. For all $1\le i\le n$, $O_i$ is the circumcenter of triangle $A_{i-1}A_iA_{i+1}$ (where $A_0=A_n$ and $A_{n+1}=A_1$). Prove that the closed path $O_1O_2\dots O_n$ doesn't form a convex $n$-gon.
We have two concentric circles. A polygon is circumscribed around the smaller circle and is contained entirely inside the greater circle. Perpendiculars from the common center of the circles to the sides of the polygon are extended till they intersect the greater circle. Each of the points obtained is connected with the endpoints of the corresponding side of the polygon . When is the resulting star-shaped polygon the unfolding of a pyramid?
Let $A_1A_2...A_n$ ba a polygon. Prove that there is a convex polygon $B_1B_2...B_n$ such that $B_iB_{i + 1} = A_iA_{i + 1}$ for $i \in \{1, 2,...,n-1\}$ and $B_nB_1 = A_nA_1$ (some of the successive vertices of the polygon $B_1B_2...B_n$ can be colinear).
A circle of length $10$ is inscribed in a convex polygon with perimeter $15$. What part of the area of this polygon is occupied by the resulting circle?
Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles.
[i]Author: Vyacheslav Yasinskiy, Ukraine[/i]
Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions:
[i](i)[/i] its side lengths are $ 1, 2, 3, \ldots, 1992$ in some order;
[i](ii)[/i] the polygon is circumscribable about a circle.
[i]Alternative formulation:[/i] Does there exist a 1992-gon with side lengths $ 1, 2, 3, \ldots, 1992$ circumscribed about a circle? Answer the same question for a 1990-gon.
$(NET 1)$ The vertices of an $(n + 1)-$gon are placed on the edges of a regular $n-$gon so that the perimeter of the $n-$gon is divided into equal parts. How does one choose these $n + 1$ points in order to obtain the $(n + 1)-$gon with
$(a)$ maximal area;
$(b)$ minimal area?
Find all values of positive integers $a$ and $b$ such that it is possible to put $a$ ones and $b$ zeros in every of vertices in polygon with $a+b$ sides so it is possible to rotate numbers in those vertices with respect to primary position and after rotation one neighboring $0$ and $1$ switch places and in every other vertices other than those two numbers remain the same.
All diagonals of a convex polygon are drawn. Prove that its sides and diagonals can be assigned arrows in such a way that no round trip along sides and diagonals is possible.
On the sides of a convex, non-regular $m$-gon are built externally regular heptagons. It is known that their centers are vertices of a regular $m$-gon. What’s the least possible value of $m$?
Let $ABCD \dots KLMN$ be a regular polygon with $14$ sides. Show that the diagonals $AE$, $BG$, and $CK$ are concurrent.
Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles.
[i]Author: Vyacheslav Yasinskiy, Ukraine[/i]
Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.
Let $ABCDEFGHIJ$ be a regular $10$-sided polygon that has all its vertices in one circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at $P$ and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the measure of the segment $PQ$.