Olympiad problems

1982 IMO Longlists, 41

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.

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?

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.

2023 OMpD, 2

Tags: geometry , pentagon , convex polygon , inscribed figures , area

Let $ABCDE$ be a convex pentagon inscribed in a circle $\Gamma$, such that $AB = BC = CD$. Let $F$ and $G$ be the intersections of $BE$ with $AC$ and of $CE$ with $BD$, respectively. Show that: a) $[ABC] = [FBCG]$ b) $\frac{[EFG]}{[EAD]} = \frac{BC}{AD}$ [b]Note: [/b] $[X]$ denotes the area of polygon $X$.

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$.

2010 Sharygin Geometry Olympiad, 3

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

All sides of a convex polygon were decreased in such a way that they formed a new convex polygon. Is it possible that all diagonals were increased?

1954 Moscow Mathematical Olympiad, 259

Tags: geometry , pentagon , convex polygon

A regular star-shaped hexagon is split into $4$ parts. Construct from them a convex polygon. Note: A regular six-pointed star is a figure that is obtained by combining a regular triangle and a triangle symmetrical to it relative to its center

1977 Chisinau City MO, 146

Tags: convex , polygon , 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.

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.

1989 IMO Shortlist, 18

Tags: geometry , algebra , convex polygon , area

Given a convex polygon $ A_1A_2 \ldots A_n$ with area $ S$ and a point $ M$ in the same plane, determine the area of polygon $ M_1M_2 \ldots M_n,$ where $ M_i$ is the image of $ M$ under rotation $ R^{\alpha}_{A_i}$ around $ A_i$ by $ \alpha_i, i \equal{} 1, 2, \ldots, n.$

1966 All Russian Mathematical Olympiad, 078

Tags: convex polygon , geometry , circles

Prove that you can always pose a circle of radius $S/P$ inside a convex polygon with the perimeter $P$ and area $S$.

2012 Sharygin Geometry Olympiad, 8

Tags: geometry , convex polygon , square

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)

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?

2002 Austrian-Polish Competition, 2

Tags: geometry , convex polygon , convex

Let $P_{1}P_{2}\dots P_{2n}$ be a convex polygon with an even number of corners. Prove that there exists a diagonal $P_{i}P_{j}$ which is not parallel to any side of the polygon.

2005 Sharygin Geometry Olympiad, 4

Tags: geometry , convex polygon , trigonometry

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?

2006 Junior Tuymaada Olympiad, 3

Tags: geometry , convex polygon , area of a triangle , combinatorial 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) $.

2007 IMAC Arhimede, 6

Tags: geometry , convex , convex polygon , polygon

Let $A_1A_2...A_n$ ba a polygon. Prove that there is a convex polygon $B_1B_2...B_n$ such that $B_iB_{i + 1} = A_iA_{i + 1}$ for $i \in \{1, 2,...,n-1\}$ and $B_nB_1 = A_nA_1$ (some of the successive vertices of the polygon $B_1B_2...B_n$ can be colinear).

1994 Spain Mathematical Olympiad, 6

Tags: combinatorics , convex polygon

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 $n$.

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.

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)

1982 IMO Shortlist, 2

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

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 $4.$

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.

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.

2009 Bulgaria National Olympiad, 5

Tags: combinatorics , convex polygon , coloring , color problem , diagonal

We divide a convex $2009$-gon in triangles using non-intersecting diagonals. One of these diagonals is colored green. It is allowed the following operation: for two triangles $ABC$ and $BCD$ from the dividing/separating with a common side $BC$ if the replaced diagonal was green it loses its color and the replacing diagonal becomes green colored. Prove that if we choose any diagonal in advance it can be colored in green after applying the operation described finite number of times.

1984 IMO Shortlist, 4

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$.