This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 121

1996 Estonia National Olympiad, 3

The vertices of the quadrilateral $ABCD$ lie on a single circle. The diagonals of this rectangle divide the angles of the rectangle at vertices $A$ and $B$ and divides the angles at vertices $C$ and $D$ in a $1: 2$ ratio. Find angles of the quadrilateral $ABCD$.

2013 Oral Moscow Geometry Olympiad, 4

Similar triangles $ABM, CBP, CDL$ and $ADK$ are built on the sides of the quadrilateral $ABCD$ with perpendicular diagonals in the outer side (the neighboring ones are oriented differently). Prove that $PK = ML$.

1981 Tournament Of Towns, (009) 3

$ABCD$ is a convex quadrilateral inscribed in a circle with centre $O$, and with mutually perpendicular diagonals. Prove that the broken line $AOC$ divides the quadrilateral into two parts of equal area. (V Varvarkin)

1986 Tournament Of Towns, (110) 4

We are given the square $ABCD$. On sides $AB$ and $CD$ we are given points $ K$ and $L$ respectively, and on segment $KL$ we are given point $M$ . Prove that the second intersection point (i.e. the one other than $M$) of the intersection points of circles circumscribed around triangles $AKM$ and $MLC$ lies on the diagonal $AC$. (V . N . Dubrovskiy)

2011 Bundeswettbewerb Mathematik, 3

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

1957 Moscow Mathematical Olympiad, 352

Of all parallelograms of a given area find the one with the shortest possible longer diagonal.

Estonia Open Senior - geometry, 2000.2.4

The diagonals of the square $ABCD$ intersect at $P$ and the midpoint of the side $AB$ is $E$. Segment $ED$ intersects the diagonal $AC$ at point $F$ and segment $EC$ intersects the diagonal $BD$ at $G$. Inside the quadrilateral $EFPG$, draw a circle of radius $r$ tangent to all the sides of this quadrilateral. Prove that $r = | EF | - | FP |$.

2004 Oral Moscow Geometry Olympiad, 6

The length of each side and each non-principal diagonal of a convex hexagon does not exceed $1$. Prove that this hexagon contains a principal diagonal whose length does not exceed $\frac{2}{\sqrt3}$.

1997 Czech And Slovak Olympiad IIIA, 6

In a parallelogram $ABCD$, triangle $ABD$ is acute-angled and $\angle BAD = \pi /4$. Consider all possible choices of points $K,L,M,N$ on sides $AB,BC, CD,DA$ respectively, such that $KLMN$ is a cyclic quadrilateral whose circumradius equals those of triangles $ANK$ and $CLM$. Find the locus of the intersection of the diagonals of $KLMN$

1998 Czech And Slovak Olympiad IIIA, 5

A circle $k$ and a point $A$ outside it are given in the plane. Prove that all trapezoids, whose non-parallel sides meet at $A$, have the same intersection of diagonals.

1966 IMO Longlists, 53

Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$

2009 Bundeswettbewerb Mathematik, 4

How many diagonals can you draw in a convex $2009$-gon if in the finished drawing, every drawn diagonal inside the $2009$-gon may cut at most another drawn diagonal?

2014 Oral Moscow Geometry Olympiad, 3

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

2006 Sharygin Geometry Olympiad, 8.5

Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?

1950 Moscow Mathematical Olympiad, 181

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?

1986 All Soviet Union Mathematical Olympiad, 433

Find the relation of the black part length and the white part length for the main diagonal of the a) $100\times 99$ chess-board; b) $101\times 99$ chess-board.

2008 Hanoi Open Mathematics Competitions, 8

The sides of a rhombus have length $a$ and the area is $S$. What is the length of the shorter diagonal?

2006 Germany Team Selection Test, 3

Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$. [i]Proposed by Alexander Ivanov, Bulgaria[/i]

1975 Bundeswettbewerb Mathematik, 3

Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.

2009 Bulgaria National Olympiad, 5

We divide a convex $2009$-gon in triangles using non-intersecting diagonals. One of these diagonals is colored green. It is allowed the following operation: for two triangles $ABC$ and $BCD$ from the dividing/separating with a common side $BC$ if the replaced diagonal was green it loses its color and the replacing diagonal becomes green colored. Prove that if we choose any diagonal in advance it can be colored in green after applying the operation described finite number of times.

2012 Sharygin Geometry Olympiad, 7

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)

2001 239 Open Mathematical Olympiad, 2

In a convex quadrangle $ ABCD $, the rays $ DA $ and $ CB $ intersect at point $ Q $, and the rays $ BA $ and $ CD $ at the point $ P $. It turned out that $ \angle AQB = \angle APD $. The bisectors of the angles $ \angle AQB $ and $ \angle APD $ intersect the sides quadrangle at points $ X $, $ Y $ and $ Z $, $ T $ respectively. Circumscribed circles of triangles $ ZQT $ and $ XPY $ intersect at $ K $ inside quadrangle. Prove that $ K $ lies on the diagonal $ AC $.

2003 Junior Balkan Team Selection Tests - Moldova, 3

Tags: diagonal , ratio , area , geometry
The quadrilateral $ABCD$ with perpendicular diagonals is inscribed in the circle with center $O$, the points $M,N$ are the midpoints of $[BC]$ and $[CD]$ respectively. Find the ratio of areas of the figures $OMCN$ and $ABCD$

1957 Moscow Mathematical Olympiad, 346

Find all isosceles trapezoids that are divided into $2$ isosceles triangles by a diagonal.

2018 Oral Moscow Geometry Olympiad, 1

Two parallelograms are arranged so as it shown on the picture. Prove that the diagonal of the one parallelogram passes through the intersection point of the diagonals of the second. [img]https://cdn.artofproblemsolving.com/attachments/9/a/15c2f33ee70eec1bcc44f94ec0e809c9e837ff.png[/img]