Found problems: 200
Let $A_1A_2\dots A_{11}$ be a non-convex $11$-gon such that
- The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$,
- $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$,
- The perimeter of $A_1A_2\dots A_{11}$ is $20$.
If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive integers $m,n,p,q$ with $n$ squarefree and $\gcd(m,p,q)=1$, find $m+n+p+q$.
All diagonals are plotted in a $2017$-sided convex polygon. A line $\ell$ intersects said polygon but does not pass through any of its vertices. Show that the line $\ell$ intersects an even number of diagonals of said polygon.
Find the number of sequences of $0, 1$ with length $n$ satisfying both of the following properties:
[list]
[*] There exists a simple polygon such that its $i$-th angle is less than $180$ degrees if and only if the $i$-th element of the sequence is $1$.
[*] There exists a convex polygon such that its $i$-th angle is less than $90$ degrees if and only if the $i$-th element of the sequence is $1$.
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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 $PQ$.
A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.
A $101$-gon is inscribed in a circle. From each vertex of this polygon a perpendicular is dropped to the opposite side or its extension. Prove that at least one perpendicular drops to the side.
A broken line $A_1A_2 \ldots A_n$ is drawn in a $50 \times 50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248.$
Given right $n$-gon wit the point $O$ -- its centre. All the vertices are marked either with $+1$ or $-1$. We may change all the signs in the vertices of regular $k$-gon ($2 \le k \le n$) with the same centre $O$. (By $2$-gon we understand a segment, being halved by $O$.) Prove that in a), b) and c) cases there exists such a set of $(+1)$s and $(-1)$s, that we can never obtain a set of $(+1)$s only.
a) $n = 15$,
b) $n = 30$,
c) $n > 2$,
d) Let us denote $K(n)$ the maximal number of $(+1)$ and $(-1)$ sets such, that it is impossible to obtain one set from another. Prove, for example, that $K(200) = 2^{80}$
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
Can four equal polygons be placed on the plane in such a way that any two of them don't have common interior points, but have a common boundary segment?
(S.Markelov)
Suppose that the vertices of a regular polygon of $20$ sides are coloured with three colours - red, blue and green - such that there are exactly three red vertices. Prove that there are three vertices $A,B,C$ of the polygon having the same colour such that triangle $ABC$ is isosceles.
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?
Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.
Let $C$ be a coloring of all edges and diagonals of a convex $n$−gon in red and blue (in Romanian, rosu and albastru). Denote by $q_r(C)$ (resp. $q_a(C)$) the number of quadrilaterals having all its edges and diagonals red (resp. blue).
Prove: $ \underset{C}{min} (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}$
Is there a $1987$-gon with consecutive sides lengths $1, 2, 3,..., 1986, 1987$, in which you can fit a circle?
Each of two regular polygons $P$ and $Q$ was divided by a line into two parts. One part of $P$ was attached to one part of $Q$ along the dividing line so that the resulting polygon was regular and not congruent to $P$ or $Q$. How many sides can it have?
$(FRA 3)$ 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 $T = 2S- B -2I + 2.$
a) What can be the number of symmetry axes of a checked polygon, that is, of a polygon whose sides lie on lines of a list of checked paper? (Indicate all possible values.)
b) What can be the number of symmetry axes of a checked polyhedron, that is, of a polyhedron consisting of equal cubes which border one to another by plane facets?
You should paint all the sides and diagonals of the regular $n$-gon so, that every pair of segments, having the common point, would be painted with different colours. How many colours will you require?
How many colorings of an $n$-gon in $p \ge 2$ colors are there such that no two neighboring vertices have the same color?
What is a centrally symmetric polygon of greatest area one can inscribe in a given triangle?
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.
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]
A square is given. Lines divide it into $n$ polygons.
What is he the largest possible sum of the internal angles of all polygons?
Prove that every convex polygon has at most one triangulation consisting entirely of acute triangles.