Olympiad problems

1955 Moscow Mathematical Olympiad, 295

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.

1974 All Soviet Union Mathematical Olympiad, 191

Tags: diagonal , hexagon , convex , geometry

a) Each of the side of the convex hexagon is longer than $1$. Does it necessary have a diagonal longer than $2$? b) Each of the main diagonals of the convex hexagon is longer than $2$. Does it necessary have a side longer than $1$?

1986 Czech And Slovak Olympiad IIIA, 4

Tags: convex , set , subset , combinatorics

Let $C_1,C_2$, and $C_3$ be points inside a bounded convex planar set $M$. Rays $l_1,l_2,l_3$ emanating from $C_1,C_2,C_3$ respectively partition the complement of the set $M \cup l_1 \cup l_2 \cup l_3$ into three regions $D_1,D_2,D_3$. Prove that if the convex sets $A$ and $B$ satisfy $A\cap l_j =\emptyset = B\cap l_j$ and $A\cap D_j \ne \emptyset \ne B\cap D_j$ for $j = 1,2,3$, then $A\cap B \ne \emptyset$

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?

2016 Saudi Arabia IMO TST, 2

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

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

2014 Contests, 3

Tags: convex , diagonal , pentagon , geometry

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

2015 Iran Geometry Olympiad, 1

Tags: convex , elementary , geometry

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]

2001 Estonia National Olympiad, 1

Tags: geometry , angle , convex , polygon

A convex $n$-gon has exactly three obtuse interior angles. Find all possible values of $n$.

1977 Chisinau City MO, 146

Tags: polygon , convex , 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.

1948 Moscow Mathematical Olympiad, 151

Tags: midpoint , distance , convex , parallel , geometry

The distance between the midpoints of the opposite sides of a convex quadrilateral is equal to a half sum of lengths of the other two sides. Prove that the first pair of sides is parallel.

1987 All Soviet Union Mathematical Olympiad, 458

Tags: area , diagonal , convex , pentagon

The convex $n$-gon ($n\ge 5$) is cut along all its diagonals. Prove that there are at least a pair of parts with the different areas.

1985 Tournament Of Towns, (104) 1

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

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)

1947 Moscow Mathematical Olympiad, 136

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

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

2011 QEDMO 9th, 5

Tags: perpendicular , convex polygon , convex , geometry

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 $[AB]$.

2005 Estonia National Olympiad, 2

Tags: polygon , convex , equal angles , 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 $\angle AOB =\gamma$ .)

2002 Regional Competition For Advanced Students, 3

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

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.

1995 Poland - Second Round, 2

Tags: concurrency , hexagon , convex , geometry

Let $ABCDEF$ be a convex hexagon with $AB = BC, CD = DE$ and $EF = FA$. Prove that the lines through $C,E,A$ perpendicular to $BD,DF,FB$ are concurrent.

1962 Kurschak Competition, 2

Tags: geometry , convex , diagonal

Show that given any $n+1$ diagonals of a convex $n$-gon, one can always find two which have no common point.

2011 Tournament of Towns, 3

Tags: angle , geometry , convex , quadrilateral

In a convex quadrilateral $ABCD, AB = 10, BC = 14, CD = 11$ and $DA = 5$. Determine the angle between its diagonals.

1984 Tournament Of Towns, (054) O2

Tags: equal segments , convex , pentagon , geometry

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.

1988 All Soviet Union Mathematical Olympiad, 464

Tags: geometry , equal area , convex , area

$ABCD$ is a convex quadrilateral. The midpoints of the diagonals and the midpoints of $AB$ and $CD$ form another convex quadrilateral $Q$. The midpoints of the diagonals and the midpoints of $BC$ and $CA$ form a third convex quadrilateral $Q'$. The areas of $Q$ and $Q'$ are equal. Show that either $AC$ or $BD$ divides $ABCD$ into two parts of equal area.

1978 Chisinau City MO, 168

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

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

1999 German National Olympiad, 4

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

A convex polygon $P$ is placed inside a unit square $Q$. Prove that the perimeter of $P$ does not exceed $4$.

Ukrainian TYM Qualifying - geometry, IV.8

Tags: geometry , hexagon , convex , diagonal , area , geometric inequality

Prove that in an arbitrary convex hexagon there is a diagonal that cuts off from it a triangle whose area does not exceed $\frac16$ of the area of the hexagon. What are the properties of a convex hexagon, each diagonal of which is cut off from it is a triangle whose area is not less than $\frac16$ the area of the hexagon?