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: 86

2009 Balkan MO Shortlist, G3
Tags: diagonals , projections , geometry , midpoints , convex quadrilateral , Circumcenter , Isogonal conjugate
Let $ABCD$ be a convex quadrilateral, and $P$ be a point in its interior. The projections of $P$ on the sides of the quadrilateral lie on a circle with center $O$. Show that $O$ lies on the line through the midpoints of $AC$ and $BD$.

2005 Slovenia Team Selection Test, 1
Tags: equal angles , geometry , diagonals
The diagonals of a convex quadrilateral $ABCD$ intersect at $M$. The bisector of $\angle ACD$ intersects the ray $BA$ at $K$. Prove that if $MA\cdot MC + MA\cdot CD = MB \cdot MD $, then $\angle BKC = \angle BDC$

2012 Estonia Team Selection Test, 3
Tags: cyclic quadrilateral , acute , diagonals , geometry , circumcircle
In a cyclic quadrilateral $ABCD$ we have $|AD| > |BC|$ and the vertices $C$ and $D$ lie on the shorter arc $AB$ of the circumcircle. Rays $AD$ and $BC$ intersect at point $K$, diagonals $AC$ and $BD$ intersect at point $P$. Line $KP$ intersects the side $AB$ at point $L$. Prove that $\angle ALK$ is acute.

2014 Contests, 3
Tags: geometry , perpendicular , diagonals
Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?

2017 Czech-Polish-Slovak Junior Match, 2
Tags: hexagon , diagonals , geometry
Decide if exists a convex hexagon with all sides longer than $1$ and all nine of its diagonals are less than $2$ in length.

1981 Tournament Of Towns, (009) 3
Tags: geometry , Cyclic , perpendicular , diagonals , area , bisect area
$ABCD$ 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)

1994 Czech And Slovak Olympiad IIIA, 3
Tags: combinatorial geometry , combinatorics , diagonals
A convex $1994$-gon $M$ is given in the plane. A closed polygonal line consists of $997$ of its diagonals. Every vertex is adjacent to exactly one diagonal. Each diagonal divides $M$ into two sides, and the smaller of the numbers of edges on the two sides of $M$ is defined to be the length of the diagonal. Is it posible to have (a) $991$ diagonals of length $3$ and $6$ of length $2$? (b) $985$ diagonals of length $6, 4$ of length $8$, and $8$ of length $3$?

2011 Abels Math Contest (Norwegian MO), 2b
Tags: geometry , hexagon , diagonals , concurrency , concurrent , area of a triangle , areas
The diagonals $AD, BE$, and $CF$ of a convex hexagon $ABCDEF$ intersect in a common point. Show that $a(ABE) a(CDA) a(EFC) = a(BCE) a(DEA) a(FAC)$, where $a(KLM)$ is the area of the triangle $KLM$. [img]https://cdn.artofproblemsolving.com/attachments/0/a/bcbbddedde159150fe3c26b1f0a2bfc322aa1a.png[/img]

2011 Sharygin Geometry Olympiad, 1
Tags: geometry , trapezoid , diagonals , perpendicular , isosceles
The diagonals of a trapezoid are perpendicular, and its altitude is equal to the medial line. Prove that this trapezoid is isosceles

2005 Sharygin Geometry Olympiad, 11.3
Tags: ratio , distance , cyclic quadrilateral , geometry , diagonals
Inside the inscribed quadrilateral $ABCD$ there is a point $K$, the distances from which to the sides $ABCD$ are proportional to these sides. Prove that $K$ is the intersection point of the diagonals of $ABCD$.

2009 Oral Moscow Geometry Olympiad, 1
Tags: geometry , sidelengths , diagonals
Are there two such quadrangles that the sides of the first are less than the corresponding sides of the second, and the corresponding diagonals are larger? (Arseniy Akopyan)

2014 India PRMO, 3
Tags: geometry , perpendicular , diagonals
Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?

2015 Sharygin Geometry Olympiad, 6
Tags: geometry , concurrency , concurrent , diagonals
The diagonals of convex quadrilateral $ABCD$ are perpendicular. Points $A' , B' , C' , D' $ are the circumcenters of triangles $ABD, BCA, CDB, DAC$ respectively. Prove that lines $AA' , BB' , CC' , DD' $ concur. (A. Zaslavsky)

2011 Sharygin Geometry Olympiad, 8
Tags: geometry , diagonals , convex polygon , circumcircle
A convex $n$-gon $P$, where $n > 3$, is dissected into equal triangles by diagonals non-intersecting inside it. Which values of $n$ are possible, if $P$ is circumscribed?

1997 Nordic, 2
Tags: geometry , area of triangle , diagonals , area , quadrilateral
Let $ABCD$ be a convex quadrilateral. We assume that there exists a point $P$ inside the quadrilateral such that the areas of the triangles $ABP, BCP, CDP$, and $DAP$ are equal. Show that at least one of the diagonals of the quadrilateral bisects the other diagonal.

2004 Tournament Of Towns, 3
Tags: geometry , perimeter , diagonals
Perimeter of a convex quadrilateral is $2004$ and one of its diagonals is $1001$. Can another diagonal be $1$ ? $2$ ? $1001$ ?

2009 Chile National Olympiad, 2
Tags: geometry , diagonals , diagonal
Consider $P$ a regular $9$-sided convex polygon with each side of length $1$. A diagonal at $P$ is any line joining two non-adjacent vertices of $P$. Calculate the difference between the lengths of the largest and smallest diagonal of $P$.

1998 Czech And Slovak Olympiad IIIA, 5
Tags: geometry , trapezoid , Fixed point , diagonals
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.

2012 Sharygin Geometry Olympiad, 4
Tags: geometry , regular polygon , diagonals
Determine all integer $n > 3$ for which a regular $n$-gon can be divided into equal triangles by several (possibly intersecting) diagonals. (B.Frenkin)

2019 Oral Moscow Geometry Olympiad, 2
Tags: congruent , quadrilateral , diagonals , congruence , geometry
The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily similar?

2003 Junior Balkan Team Selection Tests - Moldova, 3
Tags: ratio , geometry , areas , diagonals
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 $ABCD$

2006 Sharygin Geometry Olympiad, 10
Tags: diagonals , regular polygon , cut , isosceles , combinatorial geometry , geometry
At what $n$ can a regular $n$-gon be cut by disjoint diagonals into $n- 2$ isosceles (including equilateral) triangles?

2016 Oral Moscow Geometry Olympiad, 1
Tags: hexagon , angles , diagonals , equal segments , geometry
Angles are equal in a hexagon, three main diagonals are equal and the other six diagonals are also equal. Is it true that it has equal sides?

1998 Tournament Of Towns, 4
Tags: combinatorial geometry , combinatorics , diagonals , concurrent
All the diagonals of a regular $25$-gon are drawn. Prove that no $9$ of the diagonals pass through one interior point of the $25$-gon. (A Shapovalov)

2018 Oral Moscow Geometry Olympiad, 2
Tags: geometry , trapezoid , perpendicular , diagonals , right angle
The diagonals of the trapezoid $ABCD$ are perpendicular ($AD//BC, AD>BC$) . Point $M$ is the midpoint of the side of $AB$, the point $N$ is symmetric of the center of the circumscribed circle of the triangle $ABD$ wrt $AD$. Prove that $\angle CMN = 90^o$. (A. Mudgal, India)