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

2007 Hanoi Open Mathematics Competitions, 3
Tags: geometry , diagonal
Which of the following is a possible number of diagonals of a convex polygon? (A) $02$ (B) $21$ (C) $32$ (D) $54$ (E) $63$

1988 Tournament Of Towns, (176) 2
Tags: isosceles trapezoid , trapezoid , geometry , parallel , diagonal
Two isosceles trapezoids are inscribed in a circle in such a way that each side of each trapezoid is parallel to a certain side of the other trapezoid . Prove that the diagonals of one trapezoid are equal to the diagonals of the other.

Brazil L2 Finals (OBM) - geometry, 2010.5
Tags: geometry , circumcircle , cyclic quadrilateral , cyclic , diagonal
The diagonals of an cyclic quadrilateral $ABCD$ intersect at $O$. The circumcircles of triangle $AOB$ and $COD$ intersect lines $BC$ and $AD$, for the second time, at points $M, N, P$and $Q$. Prove that the $MNPQ$ quadrilateral is inscribed in a circle of center $O$.

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

2010 Sharygin Geometry Olympiad, 3
Tags: geometry , convex polygon , inequality , geometric inequality , diagonal
All sides of a convex polygon were decreased in such a way that they formed a new convex polygon. Is it possible that all diagonals were increased?

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

2021 Polish Junior MO First Round, 6
Tags: combinatorial geometry , diagonal , combinatorics
In the convex $(2n+2) $-gon are drawn $n^2$ diagonals. Prove that one of these of diagonals cuts the $(2n+2)$ -gon into two polygons, each of which has an odd number vertices.

2007 Sharygin Geometry Olympiad, 2
Tags: geometry , quadrilateral , isosceles , isosceles triangle , diagonal
Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?

Estonia Open Senior - geometry, 2000.2.4
Tags: tangential , inradius , geometry , square , midpoint , incircle , diagonal
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 $r = | EF | - | FP |$.

1966 IMO Longlists, 53
Tags: geometry , combinatorics , hexagon , diagonal , area
Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$

2011 Tournament of Towns, 4
Tags: isosceles , geometry , square , diagonal
Each diagonal of a convex quadrilateral divides it into two isosceles triangles. The two diagonals of the same quadrilateral divide it into four isosceles triangles. Must this quadrilateral be a square?

1988 Tournament Of Towns, (166) 3
Tags: combinatorial geometry , combinatorics , regular polygon , coloring , diagonal , game strategy , game
(a) The vertices of a regular $10$-gon are painted in turn black and white. Two people play the following game . Each in turn draws a diagonal connecting two vertices of the same colour . These diagonals must not intersect . The winner is the player who is able to make the last move. Who will win if both players adopt the best strategy? (b) Answer the same question for the regular $12$-gon . (V.G. Ivanov)

2007 Sharygin Geometry Olympiad, 14
Tags: equal angles , trapezoid , diagonal , geometry
In a trapezium with bases $AD$ and $BC$, let $P$ and $Q$ be the middles of diagonals $AC$ and $BD$ respectively. Prove that if $\angle DAQ = \angle CAB$ then $\angle PBA = \angle DBC$.

2005 Sharygin Geometry Olympiad, 9.1
Tags: cyclic quadrilateral , equal angles , perpendicular , geometry , diagonal
The quadrangle $ABCD$ is inscribed in a circle whose center $O$ lies inside it. Prove that if $\angle BAO = \angle DAC$, then the diagonals of the quadrilateral are perpendicular.

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 $DA$?

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

2017 Yasinsky Geometry Olympiad, 4
Tags: trapezium , trapezoid , diagonal , geometry
Diagonals of trapezium $ABCD$ are mutually perpendicular and the midline of the trapezium is $5$. Find the length of the segment that connects the midpoints of the bases of the trapezium.

1986 Tournament Of Towns, (110) 4
Tags: circumcircle , square , diagonal , geometry
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)

2004 Oral Moscow Geometry Olympiad, 6
Tags: geometry , geometric inequality , diagonal , hexagon
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 $\frac{2}{\sqrt3}$.

2004 Federal Competition For Advanced Students, P2, 3
Tags: geometry , trapezoid , area , perpendicular , circles , diagonal
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 $k_c$.

2017 Korea National Olympiad, problem 1
Tags: combinatorics , polygon , counting , diagonal
Denote $U$ as the set of $20$ diagonals of the regular polygon $P_1P_2P_3P_4P_5P_6P_7P_8$. Find the number of sets $S$ which satisfies the following conditions. 1. $S$ is a subset of $U$. 2. If $P_iP_j \in S$ and $P_j P_k \in S$, and $i \neq k$, $P_iP_k \in S$.

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)

2009 Bulgaria National Olympiad, 5
Tags: combinatorics , convex polygon , coloring , color problem , diagonal
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.

2009 Postal Coaching, 2
Tags: max , area , polygon , cyclic , perpendicular , diagonal , geometry
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.

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]