Olympiad problems

1986 IMO Shortlist, 16

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 $X$.

1998 Israel National Olympiad, 7

Tags: geometry , polygon , parallel , combinatorial geometry , combinatorics

A polygonal line of the length $1001$ is given in a unit square. Prove that there exists a line parallel to one of the sides of the square that meets the polygonal line in at least $500$ points.

1977 IMO Longlists, 29

Tags: complex number , geometry , square , polygon

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.

1989 IMO Longlists, 16

Tags: geometry , angle , geometric inequality , polygon

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.$

1992 IMO Shortlist, 8

Tags: algebra , convex polygon , incircle , polygon , additive combinatorics

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.

1966 IMO Longlists, 41

Tags: geometry , polygon , counting , obtuse triangle

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 ?

2012 Czech-Polish-Slovak Junior Match, 4

Tags: polygon , regular polygon , combinatorics , combinatorial geometry , isosceles

Prove that among any $51$ vertices of the $101$-regular polygon there are three that are the vertices of an isosceles triangle.

2008 Tournament Of Towns, 4

Tags: combinatorial geometry , convex polygon , polygon , square table , geometry

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.

1969 IMO Longlists, 46

Tags: geometry , perimeter , polygon , area , optimization

$(NET 1)$ 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?

1989 Austrian-Polish Competition, 4

Tags: convex , polygon , geometry , combinatorial geometry , circles

Let $P$ be a convex polygon in the plane. Show that there exists a circle containing the entire polygon $P$ and having at least three adjacent vertices of $P$ on its boundary.

May Olympiad L1 - geometry, 2018.3

Tags: computational , polygon , geometry

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 $PQ$.

1971 IMO Shortlist, 14

Tags: combinatorics , length , point set , polygon

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 $1248.$

1970 IMO Shortlist, 1

Tags: trigonometry , polygon , trigonometric identities , geometry

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 \[\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.\]

1974 Bundeswettbewerb Mathematik, 2

Tags: geometry , area , polygon , square

Seven polygons of area $1$ lie in the interior of a square with side length $2$. Show that there are two of these polygons whose intersection has an area of at least $1\slash 7.$

2013 Bosnia and Herzegovina Junior BMO TST, 4

Tags: polygon , vertices , combinatorics

It is given polygon with $2013$ sides $A_{1}A_{2}...A_{2013}$. His vertices are marked with numbers such that sum of numbers marked by any $9$ consecutive vertices is constant and its value is $300$. If we know that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$, determine with which number is marked $A_{2013}$

2007 IMO Shortlist, 8

Tags: combinatorics , combinatorial geometry , polygon , extremal combinatorics

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]

2018 Malaysia National Olympiad, A3

Tags: combinatorics , polygon

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 $n$?

2002 Spain Mathematical Olympiad, Problem 6

Tags: polygon , coloring , geometry

In a regular polygon $H$ of $6n+1$ sides ($n$ is a positive integer), we paint $r$ vertices red, and the rest blue. Demonstrate that the number of isosceles triangles that have three of their vertices of the same color does not depend on the way we distribute the colors on the vertices of $H$.

2019 India PRMO, 15

Tags: parallel , polygon

In how many ways can a pair of parallel diagonals of a regular polygon of $10$ sides be selected?

2010 Germany Team Selection Test, 2

Tags: geometry , parallelogram , geometric inequality , area , polygon

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]

2001 Estonia National Olympiad, 1

Tags: convex , polygon , angle , geometry

A convex $n$-gon has exactly three obtuse interior angles. Find all possible values of $n$.

1986 IMO, 1

Tags: rotation , polygon , locus , locus problems , geometry

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 $X$.

2010 IFYM, Sozopol, 7

Tags: geometry , polygon

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).

2010 Sharygin Geometry Olympiad, 7

Tags: geometry , regular polygon , paper , polygon

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?

2003 Bosnia and Herzegovina Team Selection Test, 5

Tags: polygon , quadrilateral , combinatorial geometry , combinatorics

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 $u-v$