Olympiad problems

2021 China Team Selection Test, 5

Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.

1954 Moscow Mathematical Olympiad, 265

Tags: vector , convex polygon , perpendicular

From an arbitrary point $O$ inside a convex $n$-gon, perpendiculars are drawn on (extensions of the) sides of the $n$-gon. Along each perpendicular a vector is constructed, starting from $O$, directed towards the side onto which the perpendicular is drawn, and of length equal to half the length of the corresponding side. Find the sum of these vectors.

1982 IMO Shortlist, 2

Tags: analytic geometry , area , convex polygon , geometric inequality , geometry

Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that \[\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),\] where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$

2016 IFYM, Sozopol, 3

Tags: convex polygon , rotation , 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 $c_2$.

2018 India IMO Training Camp, 3

Tags: convex polygon , combinatorics

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.

2013 Sharygin Geometry Olympiad, 3

Tags: convex polygon , projection , geometry

Each vertex of a convex polygon is projected to all nonadjacent sidelines. Can it happen that each of these projections lies outside the corresponding side?

2021 Brazil National Olympiad, 6

Tags: bicentric polygon , convex polygon , inscribed circle , circumscribed , tangent circle , geometry

Let $n \geq 5$ be integer. The convex polygon $P = A_{1} A_{2} \ldots A_{n}$ is bicentric, that is, it has an inscribed and circumscribed circle. Set $A_{i+n}=A_{i}$ to every integer $i$ (that is, all indices are taken modulo $n$). Suppose that for all $i, 1 \leq i \leq n$, the rays $A_{i-1} A_{i}$ and $A_{i+2} A_{i+1}$ meet at the point $B_{i}$. Let $\omega_{i}$ be the circumcircle of $B_{i} A_{i} A_{i+1}$. Prove that there is a circle tangent to all $n$ circles $\omega_{i}$, $1 \leq i \leq n$.

1982 IMO Shortlist, 8

Tags: geometry , symmetry , convex polygon , rectangle , circles

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure.

2005 Sharygin Geometry Olympiad, 4

Tags: convex polygon , trigonometry , geometry

At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different?

2010 Oral Moscow Geometry Olympiad, 1

Tags: combinatorial geometry , cyclic , polygon , convex polygon , combinatorics , geometry

Convex $n$-gon $P$, where $n> 3$, is cut into equal triangles by diagonals that do not intersect inside it. What are the possible values of $n$ if the $n$-gon is cyclic?

Durer Math Competition CD 1st Round - geometry, 2011.D5

Tags: convex polygon , geometry , circumcircle

Is it true that in every convex polygon $3$ adjacent vertices can be selected such that their circumcirscribed circle can cover the entire polygon?

1989 Tournament Of Towns, (241) 5

Tags: convex polygon , geometry , polygon , combinatorics , combinatorial geometry

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)

1990 IMO, 3

Tags: convex polygon , perfect square , complex number , algebra

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.

2011 Sharygin Geometry Olympiad, 8

Tags: convex polygon , diagonal , geometry , circumcircle

A convex $n$-gon $P$, where $n > 3$, is dissected into equal triangles by diagonals non-intersecting inside it. Which values of $n$ are possible, if $P$ is circumscribed?

2014 Hanoi Open Mathematics Competitions, 2

Tags: convex polygon , diagonal , geometry , combinatorial geometry

How many diagonals does $11$-sided convex polygon have?

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.

1988 IMO Shortlist, 17

Tags: convex polygon , angle , pentagon , geometry

In the convex pentagon $ ABCDE,$ the sides $ BC, CD, DE$ are equal. Moreover each diagonal of the pentagon is parallel to a side ($ AC$ is parallel to $ DE$, $ BD$ is parallel to $ AE$ etc.). Prove that $ ABCDE$ is a regular pentagon.

2012 Sharygin Geometry Olympiad, 5

Tags: convex polygon , perimeter , geometry

Do there exist a convex quadrilateral and a point $P$ inside it such that the sum of distances from $P$ to the vertices of the quadrilateral is greater than its perimeter? (A.Akopyan)

1980 IMO Shortlist, 4

Tags: polynomial , convex polygon , geometry , algebra , circumcircle

Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides \[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\] are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.

1973 All Soviet Union Mathematical Olympiad, 186

Tags: convex polygon , geometry , 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.

1990 IMO Shortlist, 16

Tags: perfect square , algebra , complex number , 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.

1990 IMO Longlists, 58

Tags: complex number , convex polygon , perfect square , algebra

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.

2012 Sharygin Geometry Olympiad, 8

Tags: square , convex polygon , geometry

A square is divided into several (greater than one) convex polygons with mutually different numbers of sides. Prove that one of these polygons is a triangle. (A.Zaslavsky)

1981 All Soviet Union Mathematical Olympiad, 320

Tags: inequalities , geometry , perimeter , convex polygon , geometric inequality

A pupil has tried to make a copy of a convex polygon, drawn inside the unit circle. He draw one side, from its end -- another, and so on. Having finished, he has noticed that the first and the last vertices do not coincide, but are situated $d$ units of length far from each other. The pupil draw angles precisely, but made relative error less than $p$ in the lengths of sides. Prove that $d < 4p$.

2004 IMO Shortlist, 6

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

Let $P$ be a convex polygon. Prove that there exists a convex hexagon that is contained in $P$ and whose area is at least $\frac34$ of the area of the polygon $P$. [i]Alternative version.[/i] Let $P$ be a convex polygon with $n\geq 6$ vertices. Prove that there exists a convex hexagon with [b]a)[/b] vertices on the sides of the polygon (or) [b]b)[/b] vertices among the vertices of the polygon such that the area of the hexagon is at least $\frac{3}{4}$ of the area of the polygon. [i]Proposed by Ben Green and Edward Crane, United Kingdom[/i]