Olympiad problems

2006 Spain Mathematical Olympiad, 3

Tags: geometry , geometric inequality , convex quadrilateral , diagonals , square roots , areas

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$. Denotes by $S_1,S_2$ and $S$ the areas of the triangles $ABE$, $CDE$ and the quadrilateral $ABCD$ respectively. Prove that $\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S}$ . When equality is reached?

2011 Abels Math Contest (Norwegian MO), 2a

Tags: geometry , perpendicular , diagonals , sidelengths

In the quadrilateral $ABCD$ the side $AB$ has length $7, BC$ length $14, CD$ length $26$, and $DA$ length $23$. Show that the diagonals are perpendicular. You may assume that the quadrilateral is convex (all internal angles are less than $180^o$).

2005 IMO Shortlist, 8

Tags: combinatorics , Extremal combinatorics , Coloring , polygon , diagonals , IMO Shortlist

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]

2007 Sharygin Geometry Olympiad, 14

Tags: geometry , equal angles , diagonals , trapezoid

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 $\angle PBA = \angle DBC$.

2010 Sharygin Geometry Olympiad, 3

Tags: geometry , convex polygon , Inequality , geometric inequality , diagonals , 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?

2015 Latvia Baltic Way TST, 4

Tags: geometry , diagonals , 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?

2005 Sharygin Geometry Olympiad, 9.1

Tags: cyclic quadrilateral , equal angles , perpendicular , diagonals , geometry

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.

Brazil L2 Finals (OBM) - geometry, 2010.5

Tags: geometry , circumcircle , diagonals , cyclic quadrilateral , Cyclic

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 $O$.

2011 District Olympiad, 2

Tags: geometry , trapezoid , perpendicular , diagonals

The isosceles trapezoid $ABCD$ has perpendicular diagonals. The parallel to the bases through the intersection point of the diagonals intersects the non-parallel sides $[BC]$ and $[AD]$ in the points $P$, respectively $R$. The point $Q$ is symmetric of the point $P$ with respect to the midpoint of the segment $[BC]$. Prove that: a) $QR = AD$, b) $QR \perp AD$.

2007 Hanoi Open Mathematics Competitions, 3

Tags: geometry , diagonals

Which of the following is a possible number of diagonals of a convex polygon? (A) $02$ (B) $21$ (C) $32$ (D) $54$ (E) $63$

1998 Tournament Of Towns, 5

Tags: Tiling , combinatorial geometry , combinatorics , Squares , diagonals

A square is divided into $25$ small squares. We draw diagonals of some of the small squares so that no two diagonals share a common point (not even a common endpoint). What is the largest possible number of diagonals that we can draw? (I Rubanov)

2022 Baltic Way, 8

Tags: combinatorics , combinatorial geometry , polygon , diagonals

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 $n$?

2011 Bundeswettbewerb Mathematik, 3

Tags: pentagon , geometry , convex , diagonals , trisector , diagonal

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

2016 Sharygin Geometry Olympiad, 7

Tags: geometry , perpendicular bisector , diagonals

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

2017 Yasinsky Geometry Olympiad, 4

Tags: geometry , trapezium , trapezoid , diagonals

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.

2008 Switzerland - Final Round, 8

Tags: geometry , hexagon , Cyclic , concurrency , concurrent , diagonals

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 $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

2015 Middle European Mathematical Olympiad, 2

Tags: combinatorics , polygon , diagonals , combinatorial geometry

Let $n\ge 3$ be an integer. An [i]inner diagonal[/i] of a [i]simple $n$-gon[/i] is a diagonal that is contained in the $n$-gon. Denote by $D(P)$ the number of all inner diagonals of a simple $n$-gon $P$ and by $D(n)$ the least possible value of $D(Q)$, where $Q$ is a simple $n$-gon. Prove that no two inner diagonals of $P$ intersect (except possibly at a common endpoint) if and only if $D(P)=D(n)$. [i]Remark:[/i] A simple $n$-gon is a non-self-intersecting polygon with $n$ vertices. A polygon is not necessarily convex.

2024 Brazil National Olympiad, 3

Tags: geometry , combinatorial geometry , angle bisector , diagonals

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.

2017 Balkan MO Shortlist, C6

Tags: combinatorics , combinatorial geometry , diagonals , minimum , Balkan MO Shortlist

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?

1997 Estonia National Olympiad, 3

Tags: diagonals , parallel , pentagon , geometry , ratio

Each diagonal of a convex pentagon is parallel to one of its sides. Prove that the ratio of the length of each diagonal to the length of the corresponding parallel side is the same, and find this ratio.

Swiss NMO - geometry, 2008.8

Tags: geometry , hexagon , Cyclic , concurrency , concurrent , diagonals

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 $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

2002 Junior Balkan Team Selection Tests - Romania, 2

Tags: fixed , geometry , area , square , diagonals

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ meet at $O$. Let $m$ be the measure of the acute angle formed by these diagonals. A variable angle $xOy$ of measure $m$ intersects the quadrilateral by a convex quadrilateral of constant area. Prove that $ABCD$ is a square.

2009 Postal Coaching, 2

Tags: geometry , max , area , polygon , Cyclic , diagonals , 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.

1993 Moldova Team Selection Test, 2

Tags: geometry , convex quadrilateral , diagonals , perpendicular lines , IMO Shortlist

A convex quadrilateral has equal diagonals. An equilateral triangle is constructed on the outside of each side of the quadrilateral. The centers of the triangles on opposite sides are joined. Show that the two joining lines are perpendicular. [i]Alternative formulation.[/i] Given a convex quadrilateral $ ABCD$ with congruent diagonals $ AC \equal{} BD.$ Four regular triangles are errected externally on its sides. Prove that the segments joining the centroids of the triangles on the opposite sides are perpendicular to each other. [i]Original formulation:[/i] Let $ ABCD$ be a convex quadrilateral such that $ AC \equal{} BD.$ Equilateral triangles are constructed on the sides of the quadrilateral. Let $ O_1,O_2,O_3,O_4$ be the centers of the triangles constructed on $ AB,BC,CD,DA$ respectively. Show that $ O_1O_3$ is perpendicular to $ O_2O_4.$

2009 Bundeswettbewerb Mathematik, 4

Tags: combinatorial geometry , combinatorics , diagonals

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?