This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 200

1989 IMO Shortlist, 7
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.$

1997 Tournament Of Towns, (561) 2
Tags: geometry , broken line , polygon , convex polygon
Which of the following statements are true? (a) If a polygon can be divided into two congruent polygons by a broken line segment, it can be divided into two congruent polygons by a straight line segment. (b) If a convex polygon can be divided into two congruent polygons by a broken line segment, it can be so divided by a straight line segment. (c) If a convex polygon can be divided into two polygons by a broken line segment, one of which can be mapped onto the other by a combination of rotations and translations, it can be so divided by a straight line segment. (S Markelov,)

1969 IMO Shortlist, 20
Tags: geometry , polygon , lattice points , area
$(FRA 3)$ 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 $T = 2S- B -2I + 2.$

1947 Putnam, A3
Tags: geometry , polygon , line
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 $P.$

1969 IMO Shortlist, 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.

1978 Chisinau City MO, 168
Tags: convex polygon , convex , polygon , combinatorial geometry , combinatorics , geometry , diagonal
Find the largest possible number of intersection points of the diagonals of a convex $n$-gon.

2021 Lotfi Zadeh Olympiad, 4
Tags: angle , 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 $1$. [/list]

1983 Polish MO Finals, 1
Tags: combinatorial geometry , combinatorics , polygon , triangle
On the plane are given a convex $n$-gon $P_1P_2....P_n$ and a point $Q$ inside it, not lying on any of its diagonals. Prove that if $n$ is even, then the number of triangles $P_iP_jP_k$ containing the point $Q$ is even.

2003 IMO Shortlist, 3
Tags: geometry , combinatorial geometry , polygon , extremal combinatorics , right angle , angle
Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles. [i]Proposed by Juozas Juvencijus Macys, Lithuania[/i]

1949 Moscow Mathematical Olympiad, 170
Tags: geometry , symmetry , inscribed , polygon
What is a centrally symmetric polygon of greatest area one can inscribe in a given triangle?

2009 Dutch IMO TST, 5
Tags: geometry , rational , polygon , equal segments , analytic geometry
Suppose that we are given an $n$-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that $n$ is even.

1989 All Soviet Union Mathematical Olympiad, 502
Tags: lattice points , polygon , rectangle , tiling
Show that for each integer $n > 0$, there is a polygon with vertices at lattice points and all sides parallel to the axes, which can be dissected into $1 \times 2$ (and / or $2 \times 1$) rectangles in exactly $n$ ways.

2016 IMAR Test, 2
Tags: combinatorial geometry , combinatorics , point , intersection , line , polygon
Given a positive integer $n$, does there exist a planar polygon and a point in its plane such that every line through that point meets the boundary of the polygon at exactly $2n$ points?

1937 Moscow Mathematical Olympiad, 037
Tags: combinatorics , combinatorial geometry , polygon
Into how many parts can a convex $n$-gon be divided by its diagonals if no three diagonals meet at one point?

2016 Peru Cono Sur TST, P2
Tags: geometry , geometric inequality , inequalities , polygon , area
Let $\omega$ be a circle. For each $n$, let $A_n$ be the area of a regular $n$-sided polygon circumscribed to $\omega$ and $B_n$ the area of a regular $n$-sided polygon inscribed in $\omega$ . Try that $3A_{2015} + B_{2015}> 4A_{4030}$

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 $m$?

1963 IMO Shortlist, 3
Tags: inequalities , geometry , polygon , geometric inequality , angle
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 $a_{1}=a_{2}= \ldots= a_{n}$.

1977 IMO, 1
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.

2018-IMOC, G1
Tags: geometry , combinatorial geometry , polygon , point
Given an integer $n \ge 3$. Find the largest positive integer $k $ with the following property: For $n$ points in general position, there exists $k$ ways to draw a non-intersecting polygon with those $n$ points as it’s vertices. [hide=Different wording]Given $n$, find the maximum $k$ so that for every general position of $n$ points , there are at least $k$ ways of connecting the points to form a polygon.[/hide]

2008 Brazil Team Selection Test, 4
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]

1967 German National Olympiad, 4
Tags: polygon , counting , obtuse triangle , geometry
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 ?

Novosibirsk Oral Geo Oly IX, 2023.5
Tags: geometry , polygon
A circle of length $10$ is inscribed in a convex polygon with perimeter $15$. What part of the area of this polygon is occupied by the resulting circle?

1998 Austrian-Polish Competition, 8
Tags: polygon , combinatorial geometry , combinatorics , grid , area
In each unit square of an infinite square grid a natural number is written. The polygons of area $n$ with sides going along the gridlines are called [i]admissible[/i], where $n > 2$ is a given natural number. The [i]value [/i] of an admissible polygon is defined as the sum of the numbers inside it. Prove that if the values of any two congruent admissible polygons are equal, then all the numbers written in the unit squares of the grid are equal. (We recall that a symmetric image of polygon $P$ is congruent to $P$.)

2012 Oral Moscow Geometry Olympiad, 2
Tags: congruent , polygon , interior , geometry
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?

1974 Polish MO Finals, 6
Tags: geometry , polygon , combinatorial geometry
Several diagonals in a convex $n$-gon are drawn so as to divide the $n$-gon into triangles and: (i) the number of diagonals drawn at each vertex is even; (ii) no two of the diagonals have a common interior point. Prove that $n$ is divisible by $3$.