Olympiad problems

2011 IMAR Test, 2

Tags: geometry , combinatorial geometry , area , lattice points , convex polygon

The area of a convex polygon in the plane is equally shared by the four standard quadrants, and all non-zero lattice points lie outside the polygon. Show that the area of the polygon is less than $4$.

2021 China Team Selection Test, 5

Tags: combinatorial geometry , symmetry , convex hull , convex polygon , area , combinatorics , geometry

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 $M$.

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.

Durer Math Competition CD 1st Round - geometry, 2018.C5

Tags: geometry , convex polygon , convex

A convex $n$-gon is called [i]nice[/i] if its sides are not all the same length, and the sum of the distances of any interior point to the side lines is $1$. Find all integers $n \ge 4$ such that a nice $n$-gon exists .

1992 IMO Longlists, 29

Tags: polygon , incircle , algebra , convex 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.

2005 Greece JBMO TST, 1

Tags: convex polygon , convex , geometry

Examine if we can place $9$ convex $6$-angled polygons the one next to the other (with common only one side or part of her) to construct a convex $39$-angled polygon.

2009 Kyiv Mathematical Festival, 4

Tags: square , perimeter , convex polygon , geometry

Two convex polygons can be placed into a square with the side $1$ without intersection. Prove that at least one polygon has the perimeter that is less than or equal to $3,5$ .

1970 All Soviet Union Mathematical Olympiad, 131

Tags: geometry , convex polygon , diagonal

How many sides of the convex polygon can equal its longest diagonal?

2006 Junior Tuymaada Olympiad, 3

Tags: combinatorial geometry , geometry , convex polygon , area of a triangle

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) $.

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.

1987 Greece National Olympiad, 1

Tags: combinatorial geometry , convex polygon , geometry

It is known that diagonals of a square, as well as a regular pentagon, are all equal. Find the bigeest natural $n$ such that a convex $n$-gon has all it's diagonals equal.

1982 IMO Longlists, 57

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

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 $4.$

1994 IMO Shortlist, 7

Tags: polygon , combinatorics , combinatorial geometry , convex polygon

Let $ n > 2$. Show that there is a set of $ 2^{n-1}$ points in the plane, no three collinear such that no $ 2n$ form a convex $ 2n$-gon.

1980 IMO, 1

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

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.

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.

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?

2003 All-Russian Olympiad Regional Round, 9.8

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

Prove that a convex polygon can be cut by disjoint diagonals into acute triangles in at least one way.

1997 Tournament Of Towns, (543) 4

Tags: geometry , convex polygon

A convex polygon $G$ is placed inside a convex polygon $ F$ so that their boundaries have no common points. A segment $s$ joining two points on the boundary of $F$ is called a support chord for $G$ if s contains a side or only a vertex of $G$. Prove that (a) there exists a support chord for $G$ such that its midpoint lies on the boundary of $G$, (b) there exist at least two such chords. (P Pushkar)