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: 96

1984 IMO, 2
Tags: algebra , geometry , convex polygon , perimeter , geometric inequality
Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

1973 All Soviet Union Mathematical Olympiad, 186
Tags: geometry , convex polygon , area
Given a convex $n$-gon with pairwise (mutually) non-parallel sides and a point inside it. Prove that there are not more than $n$ straight lines coming through that point and halving the area of the $n$-gon.

2013 China Northern MO, 1
Tags: combinatorics , combinatorial geometry , convex polygon
Find the largest positive integer $n$ ($n \ge 3$), so that there is a convex $n$-gon, the tangent of each interior angle is an integer.

2015 Sharygin Geometry Olympiad, P1
Tags: geometry , paper , convex polygon
Tanya cut out a convex polygon from the paper, fold it several times and obtained a two-layers quadrilateral. Can the cutted polygon be a heptagon?

2018 India IMO Training Camp, 3
Tags: combinatorics , convex polygon
A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the [b]interior[/b] of the polygon, in such a way that all the resulting triangles have vertices of all three colours.

1984 IMO Longlists, 33
Tags: algebra , convex polygon , perimeter , geometric inequality , geometry
Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

1990 IMO Shortlist, 16
Tags: perfect square , complex number , algebra , convex polygon
Prove that there exists a convex 1990-gon with the following two properties : [b]a.)[/b] All angles are equal. [b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.

1989 Tournament Of Towns, (241) 5
Tags: combinatorial geometry , combinatorics , geometry , convex polygon , polygon
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)

1974 All Soviet Union Mathematical Olympiad, 202
Tags: convex polygon , geometry , area
Given a convex polygon. You can put no triangle with area $1$ inside it. Prove that you can put the polygon inside a triangle with the area $4$.

2007 Sharygin Geometry Olympiad, 7
Tags: convex polygon , inscribed circle , regular polygon , geometry
A convex polygon is circumscribed around a circle. Points of contact of its sides with the circle form a polygon with the same set of angles (the order of angles may differ). Is it true that the polygon is regular?

2006 Junior Tuymaada Olympiad, 3
Tags: convex polygon , area of a triangle , combinatorial geometry , geometry
Given a convex $ n $-gon ($ n \geq 5 $). Prove that the number of triangles of area $1$ with vertices at the vertices of the $ n $-gon does not exceed $ \frac{1}{3} n (2n-5) $.

1969 Spain Mathematical Olympiad, 5
Tags: geometry , convex polygon , similarity , combinatorial geometry , tiles
Show that a convex polygon with more than four sides cannot be decomposed into two others, both similar to the first (directly or inversely), by means of a single rectilinear cut. Reasonably specify which are the quadrilaterals and triangles that admit a decomposition of this type.

2009 Tournament Of Towns, 1
Tags: convex polygon , convex , diagonal , combinatorial geometry , even , geometry
In a convex $2009$-gon, all diagonals are drawn. A line intersects the $2009$-gon but does not pass through any of its vertices. Prove that the line intersects an even number of diagonals.

1969 IMO Shortlist, 9
Tags: geometry , perimeter , rectangle , convex polygon
$(BUL 3)$ One hundred convex polygons are placed on a square with edge of length $38 cm.$ The area of each of the polygons is smaller than $\pi cm^2,$ and the perimeter of each of the polygons is smaller than $2\pi cm.$ Prove that there exists a disk with radius $1$ in the square that does not intersect any of the polygons.

1953 Moscow Mathematical Olympiad, 251
Tags: point , convex polygon , combinatorial geometry , circles
On a circle, distinct points $A_1, ... , A_{16}$ are chosen. Consider all possible convex polygons all of whose vertices are among $A_1, ... , A_{16}$ . These polygons are divided into $2$ groups, the first group comprising all polygons with $A_1$ as a vertex, the second group comprising the remaining polygons. Which group is more numerous?

1954 Moscow Mathematical Olympiad, 260
Tags: convex polygon , equal polygons , geometry
Given two convex polygons, $A_1A_2...A_n$ and $B_1B_2...B_n$ such that $A_1A_2 = B_1B_2$, $A_2A_3 = B_2B_3$,$ ...$, $A_nA_1 = B_nB_1$ and $n - 3$ angles of one polygon are equal to the respective angles of the other. Find whether these polygons are equal.

1992 IMO Longlists, 29
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.

2003 All-Russian Olympiad Regional Round, 9.8
Tags: geometry , combinatorics , combinatorial geometry , convex , convex polygon , acute
Prove that a convex polygon can be cut by disjoint diagonals into acute triangles in at least one way.

2023 Iranian Geometry Olympiad, 5
Tags: geometry , combinatorics , combinatorial geometry , triangulation , convex polygon
A polygon is decomposed into triangles by drawing some non-intersecting interior diagonals in such a way that for every pair of triangles of the triangulation sharing a common side, the sum of the angles opposite to this common side is greater than $180^o$. a) Prove that this polygon is convex. b) Prove that the circumcircle of every triangle used in the decomposition contains the entire polygon. [i]Proposed by Morteza Saghafian - Iran[/i]

2006 Sharygin Geometry Olympiad, 8.5
Tags: convex polygon , diagonal , side , geometry , convex
Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?

1990 Poland - Second Round, 6
Tags: geometry , combinatorics , combinatorial geometry , convex polygon , area
For any convex polygon $ W $ with area 1, let us denote by $ f(W) $ the area of the convex polygon whose vertices are the centers of all sides of the polygon $ W $. For each natural number $ n \geq 3 $, determine the lower limit and the upper limit of the set of numbers $ f(W) $ when $ W $ runs through the set of all $ n $ convex angles with area 1.

1994 Spain Mathematical Olympiad, 6
Tags: convex polygon , combinatorics
A convex $n$-gon is dissected into $m$ triangles such that each side of each triangle is either a side of another triangle or a side of the polygon. Prove that $m+n$ is even. Find the number of sides of the triangles inside the square and the number of vertices inside the square in terms of $m$ and $n$.

1985 Greece National Olympiad, 2
Tags: geometry , convex polygon , equilateral
a) Prove that a convex $n$-gon cannot have more than $3$ interior angles acute. b) Prove that a convex $n$-gon that has $3$ interior angles equal to $60^0,$ is equilateral.

2018 India IMO Training Camp, 3
Tags: combinatorics , convex polygon
A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the [b]interior[/b] of the polygon, in such a way that all the resulting triangles have vertices of all three colours.

2015 IFYM, Sozopol, 3
Tags: rotation , convex polygon , geometry
The angle of a rotation $\rho$ is $\alpha <180^\circ$ and $\rho$ maps the convex polygon $M$ in itself. Prove that there exist two circles $c_1$ and $c_2$ with radius $r$ and $2r$, so that $c_1$ is inner for $M$ and $M$ is inner for $c_2$.