Olympiad problems

2009 Estonia Team Selection Test, 3

Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions: (i) Each face is a regular polygon. (ii) Among the faces, there are polygons with at most two different numbers of edges. (iii) There are two faces with common edge that are both $n$-gons.

2014 Sharygin Geometry Olympiad, 3

Tags: geometry , convex , polyhedron , diagonal

Do there exist convex polyhedra with an arbitrary number of diagonals (a diagonal is a segment joining two vertices of a polyhedron and not lying on the surface of this polyhedron)? (A. Blinkov)

1985 Tournament Of Towns, (104) 1

Tags: convex , diagonal , sum , perimeter , geometric inequality

We are given a convex quadrilateral and point $M$ inside it . The perimeter of the quadrilateral has length $L$ while the lengths of the diagonals are $D_1$ and $D_2$. Prove that the sum of the distances from $M$ to the vertices of the quadrilateral are not greater than $L + D_1 + D_2$ . (V. Prasolov)

1998 Bundeswettbewerb Mathematik, 4

Tags: convex , polyhedron , combinatorial geometry , geometry

Let $3(2^n -1)$ points be selected in the interior of a polyhedron $P$ with volume $2^n$, where n is a positive integer. Prove that there exists a convex polyhedron $U$ with volume $1$, contained entirely inside $P$, which contains none of the selected points.

1978 Chisinau City MO, 168

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

Find the largest possible number of intersection points of the diagonals of a convex $n$-gon.

2004 Estonia National Olympiad, 3

Tags: geometry , bisects area , convex , parallel

On the sides $AB , BC$ of the convex quadrilateral $ABCD$ lie points $M$ and $N$ such that $AN$ and $CM$ each divide the quadrilateral $ABCD$ into two equal area parts. Prove that the line $MN$ and $AC$ are parallel.

2017 Saudi Arabia Pre-TST + Training Tests, 6

Tags: combinatorial geometry , combinatorics , convex

A convex polygon is divided into some triangles. Let $V$ and $E$ be respectively the set of vertices and the set of egdes of all triangles (each vertex in $V$ may be some vertex of the polygon or some point inside the polygon). The polygon is said to be [i]good [/i] if the following conditions hold: i. There are no $3$ vertices in $V$ which are collinear. ii. Each vertex in $V$ belongs to an even number of edges in $E$. Find all good polygon.

2015 Iran Geometry Olympiad, 1

Tags: geometry , convex , elementary

We have four wooden triangles with sides $3, 4, 5$ centimeters. How many convex polygons can we make by all of these triangles? (Just draw the polygons without any proof) A convex polygon is a polygon which all of it's angles are less than $180^o$ and there isn't any hole in it. For example: [img]https://1.bp.blogspot.com/-JgvF_B-uRag/W1R4f4AXxTI/AAAAAAAAIzc/Fo3qu3pxXcoElk01RTYJYZNwj0plJaKPQCK4BGAYYCw/s640/igo%2B2015.el1.png[/img]

2008 Swedish Mathematical Competition, 1

Tags: convex , geometry , rhombus , inscribed

A rhombus is inscribed in a convex quadrilateral. The sides of the rhombus are parallel with the diagonals of the quadrilateral, which have the lengths $d_1$ and $d_2$. Calculate the length of side of the rhombus , expressed in terms of $d_1$ and $d_2$.

1985 Tournament Of Towns, (105) 5

Tags: convex , convex polygon , acute , convex polyhedron , 3d geometry , geometry

(a) The point $O$ lies inside the convex polygon $A_1A_2A_3...A_n$ . Consider all the angles $A_iOA_j$ where $i, j$ are distinct natural numbers from $1$ to $n$ . Prove that at least $n- 1$ of these angles are not acute . (b) Same problem for a convex polyhedron with $n$ vertices. (V. Boltyanskiy, Moscow)