$(FRA 3)$ A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is $S.$ If $I$ is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number $T = 2S- B -2I + 2.$

Show that for each integer $n > 0$, there is a polygon with vertices at lattice points and all sides parallel to the axes, which can be dissected into $1 \times 2$ (and / or $2 \times 1$) rectangles in exactly $n$ ways.

No matter how two copies of a convex polygon are placed inside a square, they always have a common point. Prove that no matter how three copies of the same polygon are placed inside this square, they also have a common point.

Several diagonals in a convex $n$-gon are drawn so as to divide the $n$-gon into triangles and: (i) the number of diagonals drawn at each vertex is even; (ii) no two of the diagonals have a common interior point. Prove that $n$ is divisible by $3$.

Consider an arbitrary (optional convex) polygon. It's [i]chord [/i] is a segment whose ends lie on the boundary of the polygon, and itself belongs entirely to the polygon. Will there always be a chord of a polygon that divides it into two equal parts? Is it true that any polygon can be divided by some chord into parts, the area of each of which is not less than $\frac13$ the area of the polygon?

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

Seven congruent line segments are connected together at their endpoints as shown in the figure below at the left. By raising point $E$ the linkage can be made taller, as shown in the figure below and to the right. Continuing to raise $E$ in this manner, it is possible to use the linkage to make $A, C, F$, and $E$ collinear, while simultaneously making $B, G, D$, and $E$ collinear, thereby constructing a new triangle $ABE$. Prove that a regular polygon with center $E$ can be formed from a number of copies of this new triangle $ABE$, joined together at point $E$, and without overlapping interiors. Also find the number of sides of this polygon and justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/2/6/b3826b7ba7ea49642477878a03ac590281df43.png[/img]

Lalo and Sergio play in a regular polygon of $n\geq 4$ sides. In his turn, Lalo paints a diagonal or side of pink, and in his turn Sergio paint a diagonal or side of orange. Wins the game who achieve paint the three sides of a triangle with his color, if none of the players can win, they game tie. Lalo starts playing. Determines all natural numbers $n$ such that one of the players have winning strategy.

Convex $n$-gon $P$, where $n> 3$, is cut into equal triangles by diagonals that do not intersect inside it. What are the possible values of $n$ if the $n$-gon is cyclic?

Find all possible values of integer $n > 3$ such that there is a convex $n$-gon in which, each diagonal is the perpendicular bisector of at least one other diagonal. Proposed by Mahdi Etesamifard

Given is a regular polygon. Volodya wants to mark $k$ points on its perimeter so that any another regular polygon (maybe having a different number of sides) doesn’t contain all marked points on its perimeter. Find the minimal $k$ sufficient for any given polygon.

Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively, \[\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.\] [i]Proposed by Nairi Sedrakyan, Armenia[/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.

Two equal polygons $F$ and $F'$ are given on the plane. It is known that the vertices of the polygon $F$ belong to $F'$ (may lie inside it or on the border). Is it true that all the vertices of these polygons coincide?

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 $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon ($n \geq 3$) inscribed in a circle of radius $r$. Prove the inequality \[(n + 1)h_n+1 - nh_n > r.\] Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n \geq 3.$

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]

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 $n \geq 5$ 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]

Given an integer $n \ge 3$. Find the largest positive integer $k $ with the following property: For $n$ points in general position, there exists $k$ ways to draw a non-intersecting polygon with those $n$ points as it’s vertices. [hide=Different wording]Given $n$, find the maximum $k$ so that for every general position of $n$ points , there are at least $k$ ways of connecting the points to form a polygon.[/hide]

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]

Suppose we have two identical cardboard polygons. We placed one polygon upon the other one and aligned. Then we pierced polygons with a pin at a point. Then we turned one of the polygons around this pin by $25^o 30'$. It turned out that the polygons coincided (aligned again). What is the minimal possible number of sides of the polygons?

Given a regular polygon with $n$ sides. It is known that there are $1200$ ways to choose three of the vertices of the polygon such that they form the vertices of a [b]right triangle[/b]. What is the value of $n$?

The circle contains a closed $100$-part broken line, such that no three segments pass through one point. All its corners are obtuse, and their sum in degrees is divided by $720$. Prove that this broken line has an odd number of self-intersection points.

Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?