Olympiad problems

2012 Lusophon Mathematical Olympiad, 6

A quadrilateral $ABCD$ is inscribed in a circle of center $O$. It is known that the diagonals $AC$ and $BD$ are perpendicular. On each side we build semicircles, externally, as shown in the figure. a) Show that the triangles $AOB$ and $COD$ have the equal areas. b) If $AC=8$ cm and $BD= 6$ cm, determine the area of the shaded region.

2011 Bundeswettbewerb Mathematik, 3

Tags: pentagon , geometry , convex , 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?

1975 All Soviet Union Mathematical Olympiad, 209

Tags: hexagon , diagonal , area , geometry

Denote the midpoints of the convex hexagon $A_1A_2A_3A_4A_5A_6$ diagonals $A_6A_2$, $A_1A_3$, $A_2A_4$, $A_3A_5$, $A_4A_6$, $A_5A_1$ as $B_1, B_2, B_3, B_4, B_5, B_6$ respectively. Prove that if the hexagon $B_1B_2B_3B_4B_5B_6$ is convex, than its area equals to the quarter of the initial hexagon.

2016 Sharygin Geometry Olympiad, 7

Tags: geometry , perpendicular bisector , diagonal

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

2002 Junior Balkan Team Selection Tests - Romania, 2

Tags: fixed , geometry , area , square , diagonal

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.

1994 Czech And Slovak Olympiad IIIA, 3

Tags: combinatorial geometry , combinatorics , 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 $3$?

2009 Balkan MO Shortlist, G3

Tags: convex quadrilateral , circumcenter , isogonal conjugate , midpoint , diagonal , projection , geometry

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 $BD$.

1955 Moscow Mathematical Olympiad, 294

Tags: sum , diagonal , square table , combinatorics

a) A square table with $49$ small squares is filled with numbers $1$ to $7$ so that in each row and in each column all numbers from $1$ to $7$ are present. Let the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , 7$ are present. b) A square table with $n^2$ small squares is filled with numbers $1$ to $n$ so that in each row and in each column all numbers from $1$ to $n$ are present. Let $n$ be odd and the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , n$ are present.

2015 Oral Moscow Geometry Olympiad, 1

Tags: geometry , trapezoid , equal angles , congruent , diagonal

Two trapezoid angles and diagonals are respectively equal. Is it true that such are the trapezoid equal?

2014 Oral Moscow Geometry Olympiad, 3

Tags: geometry , convex , pentagon , diagonal

Is there a convex pentagon in which each diagonal is equal to a side?

2011 Sharygin Geometry Olympiad, 10

Tags: geometry , trapezoid , parallel , intersection diagonals , diagonal

The diagonals of trapezoid $ABCD$ meet at point $O$. Point $M$ of lateral side $CD$ and points $P, Q$ of bases $BC$ and $AD$ are such that segments $MP$ and $MQ$ are parallel to the diagonals of the trapezoid. Prove that line $PQ$ passes through point $O$.

2008 Switzerland - Final Round, 8

Tags: hexagon , cyclic , concurrency , diagonal , 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 $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

2016 Oral Moscow Geometry Olympiad, 1

Tags: hexagon , equal segments , geometry , angle , diagonal

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?

1995 Czech and Slovak Match, 5

Tags: convex quadrilateral , reflection , concyclic , orthogonal , diagonal , geometry

The diagonals of a convex quadrilateral $ABCD$ are orthogonal and intersect at point $E$. Prove that the reflections of $E$ in the sides of quadrilateral $ABCD$ lie on a circle.

1998 Tournament Of Towns, 5

Tags: tiling , combinatorial geometry , combinatorics , square , diagonal

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)

2014 Contests, 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 $DA$?

2012 Estonia Team Selection Test, 3

Tags: cyclic quadrilateral , acute , geometry , circumcircle , diagonal

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.

2018 Sharygin Geometry Olympiad, 6

Tags: incenter , excenter , geometry , circumscribed quadrilateral , diagonal

Let $ABCD$ be a circumscribed quadrilateral. Prove that the common point of the diagonals, the incenter of triangle $ABC$ and the centre of excircle of triangle $CDA$ touching the side $AC$ are collinear.

2014 Contests, 3

Tags: geometry , convex , pentagon , diagonal

Is there a convex pentagon in which each diagonal is equal to a side?

1993 Romania Team Selection Test, 3

Tags: symmetry , hexagon , geometry , diagonal , equal area

Suppose that each of the diagonals $AD,BE,CF$ divides the hexagon $ABCDEF$ into two parts of the same area and perimeter. Does the hexagon necessarily have a center of symmetry?

2009 Tournament Of Towns, 1

Tags: convex polygon , convex , diagonal , combinatorial geometry , 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.

1970 All Soviet Union Mathematical Olympiad, 131

Tags: convex polygon , geometry , diagonal

How many sides of the convex polygon can equal its longest diagonal?

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?

2006 Sharygin Geometry Olympiad, 8.5

Tags: convex polygon , diagonal , side , geometry , convex

Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?

2015 Middle European Mathematical Olympiad, 2

Tags: combinatorics , polygon , combinatorial geometry , diagonal

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.