Olympiad problems

2005 Estonia National Olympiad, 4

In a fixed plane, consider a convex quadrilateral $ABCD$. Choose a point $O$ in the plane and let $K, L, M$, and $N$ be the circumcentres of triangles $AOB, BOC, COD$, and $DOA$, respectively. Prove that there exists exactly one point $O$ in the plane such that $KLMN$ is a parallelogram.

2018 Stars of Mathematics, 4

Tags: convex polygon , disc , convex , diameter , geometric inequality , geometry

Given an integer $n \ge 3$, prove that the diameter of a convex $n$-gon (interior and boundary) containing a disc of radius $r$ is (strictly) greater than $r(1 + 1/ \cos( \pi /n))$. The Editors

2014 Contests, 3

Tags: geometry , convex , pentagon , diagonal

Is there a convex pentagon in which each diagonal is equal to a side?

2003 Belarusian National Olympiad, 8

Tags: area , convex , pentagon , geometry theorems

Given a convex pentagon $ABCDE$ with $AB=BC, CD=DE, \angle ABC=150^o, \angle CDE=30^o, BD=2$. Find the area of $ABCDE$. (I.Voronovich)

KoMaL A Problems 2018/2019, A. 737

Tags: combinatorics , graph theory , convex , polyhedron , point set , 3-dimensional geometry

$100$ points are given in space such that no four of them lie in the same plane. Consider those convex polyhedra with five vertices that have all vertices from the given set. Prove that the number of such polyhedra is even.

2005 Estonia National Olympiad, 2

Tags: equal angles , polygon , convex , geometry

Consider a convex $n$-gon in the plane with $n$ being odd. Prove that if one may find a point in the plane from which all the sides of the $n$-gon are viewed at equal angles, then this point is unique. (We say that segment $AB$ is viewed at angle $\gamma$ from point $O$ iff $\angle AOB =\gamma$ .)

1984 Tournament Of Towns, (054) O2

Tags: geometry , convex , pentagon , equal segments

In the convex pentagon $ABCDE$, $AE = AD$, $AB = AC$, and angle $CAD$ equals the sum of angles $AEB$ and $ABE$. Prove that segment $CD$ is double the length of median $AM$ of triangle $ABE$.

Ukrainian TYM Qualifying - geometry, VI.1

Tags: fixed , convex , convex quadrilateral , 3d geometry , geometry

Find all nonconvex quadrilaterals in which the sum of the distances to the lines containing the sides is the same for any interior point. Try to generalize the result in the case of an arbitrary non-convex polygon, polyhedron.

1991 Austrian-Polish Competition, 6

Tags: bisects , geometry , area of a triangle , diagonal , convex , area

Suppose that there is a point $P$ inside a convex quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal areas. Prove that one of the diagonals bisects the area of $ABCD$.

2007 IMAC Arhimede, 6

Tags: polygon , geometry , convex , convex 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).

2009 Oral Moscow Geometry Olympiad, 5

Tags: convex , geometry , polyhedron

Prove that any convex polyhedron has three edges that can be used to form a triangle. (Barbu Bercanu, Romania)

1969 Czech and Slovak Olympiad III A, 2

Tags: geometry , geometric inequalities , geometrical inequalities , convex , quadrilateral , area

Five different points $O,A,B,C,D$ are given in plane such that \[OA\le OB\le OC\le OD.\] Show that for area $P$ of any convex quadrilateral with vertices $A,B,C,D$ (not necessarily in this order) the inequality \[P\le \frac12(OA+OD)(OB+OC)\] holds and determine when equality occurs.

1987 Brazil National Olympiad, 2

Tags: geometry , convex , polyhedron

Given a point $p$ inside a convex polyhedron $P$. Show that there is a face $F$ of $P$ such that the foot of the perpendicular from $p$ to $F$ lies in the interior of $F$.

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.

2010 Mathcenter Contest, 4

Tags: geometry , functional , convex

Let $P$ be a plane. Prove that there is no function $f :P\rightarrow P$ where, for any convex quadrilateral $ABCD$, the points $f(A)$, $f(B)$, $f(C)$, $f (D)$ are the vertices of a concave quadrilateral. [i](tatari/nightmare)[/i]

1998 All-Russian Olympiad Regional Round, 10.6

Tags: geometry , pentagon , convex

The pentagon $A_1A_2A_3A_4A_5$ contains bisectors $\ell_1$, $\ell_2$, $...$, $\ell_5$ of angles $\angle A_1$, $\angle A_2$, $ ...$ , $\angle A_5$ respectively. Bisectors $\ell_1$ and $\ell_2$ intersect at point $B_1$, $\ell_2$ and $\ell_3$ - at point $B_2$, etc., $\ell_5$ and $\ell_1$ intersect at point $B_5$. Can the pentagon $B_1B_2B_3B_4B_5$ be convex?

2001 Estonia National Olympiad, 1

Tags: convex , polygon , geometry , angle

The angles of a convex $n$-gon are $a,2a, ... ,na$. Find all possible values of $n$ and the corresponding values of $a$.

1947 Moscow Mathematical Olympiad, 136

Tags: divide , 13-gon , parallelogram , convex , combinatorics

Prove that no convex $13$-gon can be cut into parallelograms.

2016 Saudi Arabia IMO TST, 2

Tags: convex , hexagon , equal segments , equal angles , geometry , angle

Let $ABCDEF$ be a convex hexagon with $AB = CD = EF$, $BC =DE = FA$ and $\angle A+\angle B = \angle C +\angle D = \angle E +\angle F$. Prove that $\angle A=\angle C=\angle E$ and $\angle B=\angle D=\angle F$. Tran Quang Hung

2008 Hanoi Open Mathematics Competitions, 7

Tags: geometry , convex , pentagon , angle

The figure $ABCDE$ is a convex pentagon. Find the sum $\angle DAC + \angle EBD +\angle ACE +\angle BDA + \angle CEB$?

1994 Tuymaada Olympiad, 4

Tags: convex , polyhedron , sphere , geometry

Let a convex polyhedron be given with volume $V$ and full surface $S$. Prove that inside a polyhedron it is possible to arrange a ball of radius $\frac{V}{S}$.

2009 Estonia Team Selection Test, 3

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

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.

1987 All Soviet Union Mathematical Olympiad, 448

Tags: geometry , combinatorics , combinatorial geometry , odd , convex , broken line

Given two closed broken lines in the plane with odd numbers of edges. All the lines, containing those edges are different, and not a triple of them intersects in one point. Prove that it is possible to chose one edge from each line such, that the chosen edges will be the opposite sides of a convex quadrangle.

2014 Sharygin Geometry Olympiad, 22

Tags: geometry , polyhedron , convex , 3-dimensional geometry

Does there exist a convex polyhedron such that it has diagonals and each of them is shorter than each of its edges?

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.