Olympiad problems

2010 Sharygin Geometry Olympiad, 7

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?

2017 Bundeswettbewerb Mathematik, 2

Tags: combinatorics , combinatorics unsolved , Coloring , Polygons , isosceles

In a convex regular $35$-gon $15$ vertices are colored in red. Are there always three red vertices that make an isosceles triangle?

1987 China Team Selection Test, 2

Tags: analytic geometry , geometry , induction , geometry unsolved , Polygons , coordinates , China

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.

2016 IFYM, Sozopol, 6

Tags: geometry , Polygons

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 $m$?

2010 IFYM, Sozopol, 5

Tags: combinatorics , polygon , Polygons

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?

2019 Auckland Mathematical Olympiad, 3

Tags: geometry , Polygons

There is a finite number of polygons in a plane and each two of them have a point in common. Prove that there exists a line which crosses every polygon.

2021 Malaysia IMONST 2, 4

Tags: geometry , Polygons

Given an octagon such that all its interior angles are equal, and all its sides have integer lengths. Prove that any pair of opposite sides have equal lengths.

1987 China Team Selection Test, 2

Tags: analytic geometry , geometry , induction , geometry unsolved , Polygons , coordinates , China

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.

2021 Lotfi Zadeh Olympiad, 4

Tags: Polygons , angles , Lotfi Zadeh MO

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 $1$. [/list]

2007 IMAC Arhimede, 6

Tags: geometry , convex , convex polygon , polygon , Polygons

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).

2005 Austrian-Polish Competition, 1

Tags: Enumeration , combinatorics , Polygons

For a convex $n$-gon $P_n$, we say that a convex quadrangle $Q$ is a [i]diagonal-quadrangle[/i] of $P_n$, if its vertices are vertices of $P_n$ and its sides are diagonals of $P_n$. Let $d_n$ be the number of diagonal-quadrangles of a convex $n$-gon. Determine $d_n$ for all $n\geq 8$.

2014 Czech-Polish-Slovak Junior Match, 5

Tags: Sum , angles , Polygons , Sqaure , combinatorial geometry , combinatorics

A square is given. Lines divide it into $n$ polygons. What is he the largest possible sum of the internal angles of all polygons?