Olympiad problems

2011 QEDMO 9th, 5

Let $P$ be a convex polygon, so have all interior angles smaller than $180^o$, and let $X$ be a point in the interior of $P$. Prove that $P$ has a side $[AB]$ such that the perpendicular from $X$ to the line $AB$ lies on the side $[AB]$.

1992 IMO Shortlist, 8

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.

2009 IMAC Arhimede, 3

Tags: geometry , convex polygon , intersection

In the interior of the convex polygon $A_1A_2...A_{2n}$ there is point $M$. Prove that at least one side of the polygon has not intersection points with the lines $MA_i$, $1\le i\le 2n$. (Spain)

1990 IMO Longlists, 58

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.

2014 Sharygin Geometry Olympiad, 8

Tags: geometry , convex polygon , minimum

A convex polygon $P$ lies on a flat wooden table. You are allowed to drive some nails into the table. The nails must not go through $P$, but they may touch its boundary. We say that a set of nails blocks $P$ if the nails make it impossible to move $P$ without lifting it off the table. What is the minimum number of nails that suffices to block any convex polygon $P$? (N. Beluhov, S. Gerdgikov)

1990 IMO, 3

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.

1966 All Russian Mathematical Olympiad, 078

Tags: convex polygon , circles , geometry

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

2014 Hanoi Open Mathematics Competitions, 2

Tags: diagonal , combinatorial geometry , geometry , convex polygon

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

1994 IMO Shortlist, 7

Tags: combinatorics , combinatorial geometry , convex polygon , 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.

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 .

Ukrainian TYM Qualifying - geometry, VI.18

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

The convex polygon $A_1A_2...A_n$ is given in the plane. Denote by $T_k$ $(k \le n)$ the convex $k$-gon of the largest area, with vertices at the points $A_1, A_2, ..., A_n$ and by $T_k(A+1)$ the convex k-gon of the largest area with vertices at the points $A_1, A_2, ..., A_n$ in which one of the vertices is in $A_1$. Set the relationship between the order of arrangement in the sequence $A_1, A_2, ..., A_n$ vertices: 1) $T_3$ and $T_3 (A_2)$ 2) $T_k$ and $T_k (A_1) $ 3) $T_k$ and $T_{k+1}$

2021 Sharygin Geometry Olympiad, 8.8

Tags: combinatorial geometry , geometry , obtuse triangle , convex polygon , equal sides

Does there exist a convex polygon such that all its sidelengths are equal and all triangle formed by its vertices are obtuse-angled?

1998 Czech and Slovak Match, 3

Tags: inequalities , convex polygon , hexagon , triangle inequality

Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE, EF = FA$. Prove that $\frac{BC}{BE} +\frac{DE}{DA} +\frac{FA}{FC} \ge \frac{3}{2}$ . When does equality occur?

2021 China Team Selection Test, 5

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

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 $M$.

1950 Moscow Mathematical Olympiad, 181

Tags: diagonal , geometry , combinatorial geometry , convex polygon

a) In a convex $13$-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have? b) In a convex $1950$-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have?

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$.

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?

1989 IMO Longlists, 62

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.$

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

Tags: geometry , circumcircle , convex polygon

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

2008 Bulgarian Autumn Math Competition, Problem 11.3

Tags: combinatorics , convex polygon , intersection diagonals

In a convex $2008$-gon some of the diagonals are coloured red and the rest blue, so that every vertex is an endpoint of a red diagonal and no three red diagonals concur at a point. It's known that every blue diagonal is intersected by a red diagonal in an interior point. Find the minimal number of intersections of red diagonals.

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)