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

2005 IMO Shortlist, 8
Tags: combinatorics , extremal combinatorics , coloring , polygon , diagonal
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]

2018 Oral Moscow Geometry Olympiad, 2
Tags: right angle , diagonal , perpendicular , trapezoid , geometry
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)

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.

2009 Postal Coaching, 2
Tags: geometry , polygon , max , cyclic , diagonal , area , perpendicular
Let $n \ge 4$ be an integer. Find the maximum value of the area of a $n$-gon which is inscribed in the circle of radius $1$ and has two perpendicular diagonals.

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.

2004 Federal Competition For Advanced Students, P2, 3
Tags: trapezoid , geometry , circles , diagonal , perpendicular , area
A trapezoid $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle $k$. Let $k_a$ and $k_c$ respectively be the circles with diameters $AB$ and $CD$. Compute the area of the region which is inside the circle $k$, but outside the circles $k_a$ and $k_c$.

2014 India PRMO, 3
Tags: geometry , perpendicular , diagonal
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 Latvia Baltic Way TST, 4
Tags: geometry , combinatorial geometry , diagonal
Can you draw some diagonals in a convex $2014$-gon so that they do not intersect, the whole $2014$-gon is divided into triangles and each vertex belongs to an odd number of these triangles?

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?

Swiss NMO - geometry, 2008.8
Tags: concurrency , cyclic , diagonal , hexagon , geometry
Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

2017 Balkan MO Shortlist, C6
Tags: combinatorics , combinatorial geometry , diagonal , minimum
What is the least positive integer $k$ such that, in every convex $101$-gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?

1978 Austrian-Polish Competition, 9
Tags: polygon , combinatorial geometry , diagonal , combinatorics
In a convex polygon $P$ some diagonals have been drawn, without intersections inside $P$. Show that there exist at least two vertices of $P$, neither one of them being an endpoint of any one of those diagonals.

2022 Baltic Way, 8
Tags: combinatorial geometry , combinatorics , diagonal , polygon
For a natural number $n \ge 3$, we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the $n$-gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of $n$?

2009 Tournament Of Towns, 1
Tags: combinatorial geometry , convex polygon , convex , diagonal , even , geometry
In a convex $2009$-gon, all diagonals are drawn. A line intersects the $2009$-gon but does not pass through any of its vertices. Prove that the line intersects an even number of diagonals.

2018 Iranian Geometry Olympiad, 3
Tags: perpendicular , geometry , diagonal , polygon
Find all possible values of integer $n > 3$ such that there is a convex $n$-gon in which, each diagonal is the perpendicular bisector of at least one other diagonal. Proposed by Mahdi Etesamifard

2022 239 Open Mathematical Olympiad, 1
Tags: combinatorics , board , diagonal , cell
A piece is placed in the lower left-corner cell of the $15 \times 15$ board. It can move to the cells that are adjacent to the sides or the corners of its current cell. It must also alternate between horizontal and diagonal moves $($the first move must be diagonal$).$ What is the maximum number of moves it can make without stepping on the same cell twice$?$

2015 Oral Moscow Geometry Olympiad, 1
Tags: equal angles , trapezoid , geometry , diagonal , congruent
Two trapezoid angles and diagonals are respectively equal. Is it true that such are the trapezoid equal?

2015 Sharygin Geometry Olympiad, 6
Tags: concurrency , diagonal , geometry
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)

2016 Oral Moscow Geometry Olympiad, 1
Tags: geometry , equal segments , angle , diagonal , hexagon
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?

2012 Estonia Team Selection Test, 3
Tags: geometry , circumcircle , cyclic quadrilateral , diagonal , acute
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.

2006 Germany Team Selection Test, 3
Tags: combinatorics , extremal combinatorics , coloring , polygon , diagonal
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]

1994 Czech And Slovak Olympiad IIIA, 3
Tags: combinatorics , combinatorial geometry , diagonal
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 $3$?

2018 Czech-Polish-Slovak Junior Match, 2
Tags: perimeter , hexagon , parallel , diagonal , geometry
A convex hexagon $ABCDEF$ is given whose sides $AB$ and $DE$ are parallel. Each of the diagonals $AD, BE, CF$ divides this hexagon into two quadrilaterals of equal perimeters. Show that these three diagonals intersect at one point.

2012 Estonia Team Selection Test, 3
Tags: circumcircle , diagonal , cyclic quadrilateral , acute , geometry
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.

2011 Sharygin Geometry Olympiad, 20
Tags: cyclic quadrilateral , tangential quadrilateral , midpoint , diagonal , geometry
Quadrilateral $ABCD$ is circumscribed around a circle with center $I$. Points $M$ and $N$ are the midpoints of diagonals $AC$ and $BD$. Prove that $ABCD$ is cyclic quadrilateral if and only if $IM : AC = IN : BD$. [i]Nikolai Beluhov and Aleksey Zaslavsky[/i]