Found problems: 200
Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that
\[\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.\]
In a convex polygon $P$ some diagonals have been drawn, without intersections inside $P$. Show that there exist at least two vertices of $P$, neither one of them being an endpoint of any one of those diagonals.
Let $F$ be a polygon the boundary of which is a broken line with vertices in the knots (units) of a given in advance regular square network. If $k$ is the count of knots of the network situated over the boundary of $F$, and $\ell$ is the count of the knots of the network lying inside $F$, prove that if the surface of every square from the network is $1$, then the surface $S$ of $F$ is calculated with the formulae:
$$S=\frac k2+\ell-1$$
[i]V. Chukanov[/i]
Let $M$ be a convex polygon. Externally, on its sides are built squares. It is known that the vertices of these squares, that don’t lie on $M$, lie on a circle $k$. Determine $M$ (its type).
Let $P$ be a convex polygon. Prove that there exists a convex hexagon that is contained in $P$ and whose area is at least $\frac34$ of the area of the polygon $P$.
[i]Alternative version.[/i] Let $P$ be a convex polygon with $n\geq 6$ vertices. Prove that there exists a convex hexagon with
[b]a)[/b] vertices on the sides of the polygon (or)
[b]b)[/b] vertices among the vertices of the polygon
such that the area of the hexagon is at least $\frac{3}{4}$ of the area of the polygon.
[i]Proposed by Ben Green and Edward Crane, United Kingdom[/i]
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]
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?
Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.
Given a triangle $ABC$ with an interior point $P$ and points $Q_1 , Q_2$ not lying on any of the segments $AB , AC ,BC,$ $AP ,BP ,CP,$ show that there does not exist a polygonal line $K$ joining $Q_1$ and $Q_2$ such that
i) $K$ crosses each segment exactly once,
ii) $K$ does not intersect itself
iii) $K$ does not pass through $A, B , C$ or $P.$
$(NET 1)$ 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?
Let $\mathbf{P}_1,\mathbf{P}_2$ be convex polygons with perimeters $o_1,o_2,$ respectively. Show that if $\mathbf P_1\subseteq\mathbf P_2,$ then $o_1\le o_2.$
A convex $n{}$-gon with $n > 4$ is such that if a diagonal cuts a triangle from it then this triangle is isosceles. Prove that there are at least 2 equal sides among any 4 sides of the $n{}$-gon.
[i]Maxim Didin[/i]
We are given a convex polygon. Show that one can find a point $Q$ inside the polygon and three vertices $A_1,A_2,A_3$ (not necessarily consecutive) such that each ray $A_iQ$ ($i=1,2,3$) makes acute angles with the two sides emanating from $A_i$.
In a convex regular $35$-gon $15$ vertices are colored in red. Are there always three red vertices that make an isosceles triangle?
Let $n$ and $c$ be coprime positive integers. For any integer $i$, denote by $i' $ the remainder of division of product $ci$ by $n$. Let $A_o.A_1,A_2,...,A_{n-1}$ be a regular $n$-gon. Prove that
a) if $A_iA_j \parallel A_kA_i$ then $A_{i'}A_{j'} \parallel A_{k'}A_{i'}$
b) if $A_iA_j \perp A_kA_l$ then $A_{i'}A_{j'} \perp A_{k'}A_{l'}$
Is there a planar polygon whose vertices have integer coordinates and whose area is $1/2$, such that this polygon is
(a) a triangle with at least two sides longer than $1000$?
(b) a triangle whose sides are all longer than $1000$?
(c) a quadrangle?
Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).
In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation
\[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \]
Prove that $a_{1}=a_{2}= \ldots= a_{n}$.
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 ?
a) Prove that if a planar polygon has several axes of symmetry, then all of them intersect at one point.
b) A finite solid body is symmetric about two distinct axes. Describe the position of the symmetry planes of the body.
Consider two triangles, $ABC$ and $DEF$, and any point $O$. We take any point $X$ in $\vartriangle ABC$ and any point $Y$ in $\vartriangle DEF$ and draw a parallelogram $OXY Z$. Prove that the locus of all possible points $Z$ form a polygon. How many sides can it have? Prove that its perimeter is equal to the sum of perimeters of the original triangles.
Let there be given a regular hexagon of side length $1$. Six circles with the sides of the hexagon as diameters are drawn. Find the area of the part of the hexagon lying outside all the circles.
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 ?
In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.
Prove that for $n\ge5$ the side of regular inscribable $n$-gon is bigger than the side of regular $n+1$-gon circumscribed around the same circle and if $n\le4$ the opposite statement is true.