Olympiad problems

2003 Belarusian National Olympiad, 8

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)

1955 Moscow Mathematical Olympiad, 295

Tags: combinatorial geometry , geometry , convex , line

Which convex domains (figures) on a plane can contain an entire straight line? It is assumed that the figure is flat and does not degenerate into a straight line and is closed, that is, it contains all its boundary points.

1989 Bundeswettbewerb Mathematik, 3

Tags: polygon , geometry , convex quadrilateral , convex

A convex polygon is divided into finitely many quadrilaterals. Prove that at least one of these quadrilaterals must also be convex.

2005 Switzerland - Final Round, 5

Tags: geometry , geometric inequalities , combinatorial geometry , convex

Tweaking a convex $n$-gon means the following: choose two adjacent sides $AB$ and $BC$ and replaces them with the line segment $AM$, $MN$, $NC$, where $M \in AB$ and $N \in BC$ are arbitrary points inside these segments. In other words, you cut off a corner and get an $(n+1)$-corner. Starting from a regular hexagon $P_6$ with area $1$, by continuous Tweaks a sequence $P_6,P_7,P_8, ...$ convex polygons. Show that Area of $P_n$ for all $n\ge 6$ greater than $\frac1 2$ is, regardless of how tweaks takes place.

2016 Saudi Arabia IMO TST, 2

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

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

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.

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.

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).

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.

2012 Sharygin Geometry Olympiad, 7

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

A convex pentagon $P $ is divided by all its diagonals into ten triangles and one smaller pentagon $P'$. Let $N$ be the sum of areas of five triangles adjacent to the sides of $P$ decreased by the area of $P'$. The same operations are performed with the pentagon $P'$, let $N'$ be the similar difference calculated for this pentagon. Prove that $N > N'$. (A.Belov)

2017 Sharygin Geometry Olympiad, P21

Tags: geometry , convex , hexagon , maximum , midpoint

A convex hexagon is circumscribed about a circle of radius $1$. Consider the three segments joining the midpoints of its opposite sides. Find the greatest real number $r$ such that the length of at least one segment is at least $r.$

2011 Bundeswettbewerb Mathematik, 3

Tags: pentagon , geometry , convex , trisector , diagonal

The diagonals of a convex pentagon divide each of its interior angles into three equal parts. Does it follow that the pentagon is regular?

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.

2005 Estonia National Olympiad, 4

Tags: parallelogram , circumcenter , convex , geometry

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.

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.

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.

1978 Swedish Mathematical Competition, 4

Tags: algebra , convex , logarithm , sequence

$b_0, b_1, b_2, \dots$ is a sequence of positive reals such that the sequence $b_0,c b_1, c^2b_2,c^3b_3,\dots$ is convex for all $c > 0$. (A sequence is convex if each term is at most the arithmetic mean of its two neighbors.) Show that $\ln b_0, \ln b_1, \ln b_2, \dots$ is convex.

1994 Tuymaada Olympiad, 4

Tags: geometry , convex , polyhedron , sphere

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 $\frac{V}{S}$.

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 .

2002 Regional Competition For Advanced Students, 3

Tags: perimeter , inequalities , convex , hexagon , geometric inequality , geometry

In the convex $ABCDEF$ (has all interior angles less than $180^o$) with the perimeter $s$ the triangles $ACE$ and $BDF$ have perimeters $u$ and $v$ respectively. a) Show the inequalities $\frac{1}{2} \le \frac{s}{u+v}\le 1$ b) Check whether $1$ is replaced by a smaller number or $1/2$ by a larger number can the inequality remains valid for all convex hexagons.

2008 Hanoi Open Mathematics Competitions, 7

Tags: geometry , convex , pentagon , angle

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

1947 Moscow Mathematical Olympiad, 126

Tags: combinatorial geometry , combinatorics , convex , pentagon , line

Given a convex pentagon $ABCDE$, prove that if an arbitrary point $M$ inside the pentagon is connected by lines with all the pentagon’s vertices, then either one or three or five of these lines cross the sides of the pentagon opposite the vertices they pass. Note: In reality, we need to exclude the points of the diagonals, because that in this case the drawn lines can pass not through the internal points of the sides, but through the vertices. But if the drawn diagonals are not considered or counted twice (because they are drawn from two vertices), then the statement remains true.

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 $\angle AOB =\gamma$ .)

2014 Oral Moscow Geometry Olympiad, 3

Tags: geometry , convex , pentagon , diagonal

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

1976 Czech and Slovak Olympiad III A, 5

Tags: geometry , perimeter , geometric inequalities , convex , polygon , inequalities

Let $\mathbf{P}_1,\mathbf{P}_2$ be convex polygons with perimeters $o_1,o_2,$ respectively. Show that if $\mathbf P_1\subseteq\mathbf P_2,$ then $o_1\le o_2.$