Olympiad problems

1989 Tournament Of Towns, (241) 5

We are given $100$ points. $N$ of these are vertices of a convex $N$-gon and the other $100 - N$ of these are inside this $N$-gon. The labels of these points make it impossible to tell whether or not they are vertices of the $N$-gon. It is known that no three points are collinear and that no $4$ points belong to two parallel lines. It has been decided to ask questions of the following type: What is the area of the triangle $XYZ$, where $X, Y$ and $Z$ are labels representing three of the $100$ given points? Prove that $300$ such questions are sufficient in order to clarify which points are vertices and to determine the area of the $N$-gon. (D. Fomin, Leningrad)

1945 Moscow Mathematical Olympiad, 100

Tags: geometry , polygon , minimum , combinatorial geometry

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?

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 BMT Spring, 11

Tags: rotation , polygon , geometry

A regular $17$-gon with vertices $V_1, V_2, . . . , V_{17}$ and sides of length $3$ has a point $ P$ on $V_1V_2$ such that $V_1P = 1$. A chord that stretches from $V_1$ to $V_2$ containing $ P$ is rotated within the interior of the heptadecagon around $V_2$ such that the chord now stretches from $V_2$ to $V_3$. The chord then hinges around $V_3$, then $V_4$, and so on, continuing until $ P$ is back at its original position. Find the total length traced by $ P$.

2016 IFYM, Sozopol, 6

Tags: geometry , polygon

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 $m$?

2001 Estonia National Olympiad, 1

Tags: geometry , angle , convex , polygon

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

2013 IFYM, Sozopol, 8

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

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]

2015 IMO Shortlist, G8

Tags: geometry , triangulation , polygon

A [i]triangulation[/i] of a convex polygon $\Pi$ is a partitioning of $\Pi$ into triangles by diagonals having no common points other than the vertices of the polygon. We say that a triangulation is a [i]Thaiangulation[/i] if all triangles in it have the same area. Prove that any two different Thaiangulations of a convex polygon $\Pi$ differ by exactly two triangles. (In other words, prove that it is possible to replace one pair of triangles in the first Thaiangulation with a different pair of triangles so as to obtain the second Thaiangulation.) [i]Proposed by Bulgaria[/i]

2000 IMO Shortlist, 4

Tags: polygon , cyclic , geometry , trigonometry

Let $ A_1A_2 \ldots A_n$ be a convex polygon, $ n \geq 4.$ Prove that $ A_1A_2 \ldots A_n$ is cyclic if and only if to each vertex $ A_j$ one can assign a pair $ (b_j, c_j)$ of real numbers, $ j = 1, 2, \ldots, n,$ so that $ A_iA_j = b_jc_i - b_ic_j$ for all $ i, j$ with $ 1 \leq i < j \leq n.$

1999 Nordic, 2

Tags: geometry , polygon , angle , maximum

Consider $7$-gons inscribed in a circle such that all sides of the $7$-gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$-gon.

1977 IMO, 1

Tags: geometry , complex number , polygon , square

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.

1977 Chisinau City MO, 146

Tags: polygon , convex , convex polygon , non-convex , geometry

Prove that $n$ ($\ge 4$) points of the plane are vertices of a convex $n$-gon if and only if any $4$ of them are vertices of a convex quadrilateral.

1982 IMO Shortlist, 6

Tags: polygon , combinatorics , length , combinatorial geometry , geometry

Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.

2013 Tournament of Towns, 5

Tags: polygon , perpendicular , geometry , combinatorial geometry

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.

1963 IMO Shortlist, 3

Tags: geometric inequality , geometry , polygon , angle , inequalities

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 $a_{1}=a_{2}= \ldots= a_{n}$.

2023 Sharygin Geometry Olympiad, 14

Tags: polygon , winding number , geometry

Suppose that a closed oriented polygonal line $\mathcal{L}$ in the plane does not pass through a point $O$, and is symmetric with respect to $O$. Prove that the winding number of $\mathcal{L}$ around $O$ is odd. The winding number of $\mathcal{L}$ around $O$ is defined to be the following sum of the oriented angles divided by $2\pi$: $$\deg_O\mathcal{L} := \dfrac{\angle A_1OA_2+\angle A_2OA_3+\dots+\angle A_{n-1}OA_n+\angle A_nOA_1}{2\pi}.$$

2021 Durer Math Competition (First Round), 3

Tags: geometry , polygon

The floor plan of a contemporary art museum is a (not necessarily convex) polygon and its walls are solid. The security guard guarding the museum has two favourite spots (points $A$ and $B$) because one can see the whole area of the museum standing at either point. Is it true that from any point of the $AB$ section one can see the whole museum?

1969 IMO Longlists, 52

Tags: polygon , dissection , geometry

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.

1974 Bundeswettbewerb Mathematik, 2

Tags: area , polygon , square , geometry

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 $1\slash 7.$

1986 IMO Shortlist, 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 $X$.

1984 All Soviet Union Mathematical Olympiad, 384

Tags: geometry , perimeter , polygon , disc , circles , area

The centre of the coin with radius $r$ is moved along some polygon with the perimeter $P$, that is circumscribed around the circle with radius $R$ ($R>r$). Find the coin trace area (a sort of polygon ring).

1989 IMO Shortlist, 7

Tags: angle , geometric inequality , polygon , geometry

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.$

2016 Ukraine Team Selection Test, 1

Tags: game , polygon , combinatorics

Consider a regular polygon $A_1A_2\ldots A_{6n+3}$. The vertices $A_{2n+1}, A_{4n+2}, A_{6n+3}$ are called [i]holes[/i]. Initially there are three pebbles in some vertices of the polygon, which are also vertices of equilateral triangle. Players $A$ and $B$ take moves in turn. In each move, starting from $A$, the player chooses pebble and puts it to the next vertex clockwise (for example, $A_2\rightarrow A_3$, $A_{6n+3}\rightarrow A_1$). Player $A$ wins if at least two pebbles lie in holes after someone's move. Does player $A$ always have winning strategy? [i]Proposed by Bohdan Rublov [/i]

2013 Bosnia and Herzegovina Junior BMO TST, 4

Tags: combinatorics , polygon , vertices

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 $A_{2013}$

2021 Malaysia IMONST 2, 4

Tags: polygon , geometry

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.