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

2016 Sharygin Geometry Olympiad, 7
Tags: diagonal , geometry , perpendicular bisector
Diagonals of a quadrilateral $ABCD$ are equal and meet at point $O$. The perpendicular bisectors to segments $AB$ and $CD$ meet at point $P$, and the perpendicular bisectors to $BC$ and $AD$ meet at point $Q$. Find angle $\angle POQ$. by A.Zaslavsky

2015 Estonia Team Selection Test, 8
Tags: diagonal , combinatorial geometry , combinatorics , regular polygon
Find all positive integers $n$ for which it is possible to partition a regular $n$-gon into triangles with diagonals not intersecting inside the $n$-gon such that at every vertex of the $n$-gon an odd number of triangles meet.

2015 Latvia Baltic Way TST, 4
Tags: diagonal , geometry , combinatorial geometry
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?

2008 Hanoi Open Mathematics Competitions, 8
Tags: geometry , rhombus , diagonal
The sides of a rhombus have length $a$ and the area is $S$. What is the length of the shorter diagonal?

2014 Hanoi Open Mathematics Competitions, 2
Tags: diagonal , combinatorial geometry , geometry , convex polygon
How many diagonals does $11$-sided convex polygon have?

1968 All Soviet Union Mathematical Olympiad, 099
Tags: geometry , diagonal
The difference between the maximal and the minimal diagonals of the regular $n$-gon equals to its side ( $n > 5$ ). Find $n$.

2009 Chile National Olympiad, 2
Tags: geometry , 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 $P$.

2024 Brazil National Olympiad, 3
Tags: diagonal , geometry , combinatorial geometry , angle bisector
Let \( n \geq 3 \) be a positive integer. In a convex polygon with \( n \) sides, all the internal bisectors of its \( n \) internal angles are drawn. Determine, as a function of \( n \), the smallest possible number of distinct lines determined by these bisectors.

2022 239 Open Mathematical Olympiad, 1
Tags: cell , combinatorics , board , diagonal
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$?$

1957 Moscow Mathematical Olympiad, 352
Tags: geometry , parallelogram , minimum , diagonal , area
Of all parallelograms of a given area find the one with the shortest possible longer diagonal.

2011 Sharygin Geometry Olympiad, 20
Tags: midpoint , diagonal , geometry , cyclic quadrilateral , tangential quadrilateral
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]

1962 Kurschak Competition, 2
Tags: diagonal , convex , geometry
Show that given any $n+1$ diagonals of a convex $n$-gon, one can always find two which have no common point.

1950 Moscow Mathematical Olympiad, 181
Tags: diagonal , geometry , combinatorial geometry , convex polygon
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?

2001 239 Open Mathematical Olympiad, 2
Tags: geometry , circumcircle , diagonal , equal angles
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 $.

2006 Sharygin Geometry Olympiad, 10
Tags: diagonal , 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?

2019 Oral Moscow Geometry Olympiad, 2
Tags: congruent , quadrilateral , congruence , diagonal , 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?

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

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

1988 Tournament Of Towns, (176) 2
Tags: diagonal , isosceles trapezoid , trapezoid , geometry , parallel
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.

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

1974 All Soviet Union Mathematical Olympiad, 191
Tags: hexagon , diagonal , geometry , convex
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 $1$?