Olympiad problems

1975 Bundeswettbewerb Mathematik, 3

Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.

1987 All Soviet Union Mathematical Olympiad, 458

Tags: convex , pentagon , area , diagonal

The convex $n$-gon ($n\ge 5$) is cut along all its diagonals. Prove that there are at least a pair of parts with the different areas.

1966 IMO Shortlist, 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 $\frac{1}{6}S.$

1950 Moscow Mathematical Olympiad, 181

Tags: geometry , combinatorial geometry , convex polygon , diagonal

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?

2009 Oral Moscow Geometry Olympiad, 1

Tags: geometry , sidelengths , diagonal

Are there two such quadrangles that the sides of the first are less than the corresponding sides of the second, and the corresponding diagonals are larger? (Arseniy Akopyan)

2022 Baltic Way, 8

Tags: combinatorial geometry , polygon , diagonal , combinatorics

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 $n$?

1995 Bundeswettbewerb Mathematik, 3

Tags: convex , pentagon , geometry , parallel , diagonal

Each diagonal of a convex pentagon is parallel to one side of the pentagon. Prove that the ratio of the length of a diagonal to that of its corresponding side is the same for all five diagonals, and compute this ratio.

2011 Sharygin Geometry Olympiad, 1

Tags: geometry , trapezoid , perpendicular , isosceles , diagonal

The diagonals of a trapezoid are perpendicular, and its altitude is equal to the medial line. Prove that this trapezoid is isosceles

2018 Czech-Polish-Slovak Junior Match, 2

Tags: hexagon , parallel , diagonal , geometry , perimeter

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.

1997 Nordic, 2

Tags: geometry , area of triangle , area , quadrilateral , diagonal

Let $ABCD$ be a convex quadrilateral. We assume that there exists a point $P$ inside the quadrilateral such that the areas of the triangles $ABP, BCP, CDP$, and $DAP$ are equal. Show that at least one of the diagonals of the quadrilateral bisects the other diagonal.

2011 Abels Math Contest (Norwegian MO), 2b

Tags: hexagon , area of a triangle , area , concurrency , diagonal , geometry

The diagonals $AD, BE$, and $CF$ of a convex hexagon $ABCDEF$ intersect in a common point. Show that $a(ABE) a(CDA) a(EFC) = a(BCE) a(DEA) a(FAC)$, where $a(KLM)$ is the area of the triangle $KLM$. [img]https://cdn.artofproblemsolving.com/attachments/0/a/bcbbddedde159150fe3c26b1f0a2bfc322aa1a.png[/img]

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.

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 $n$.

1998 Tournament Of Towns, 4

Tags: combinatorial geometry , combinatorics , concurrency , diagonal

All the diagonals of a regular $25$-gon are drawn. Prove that no $9$ of the diagonals pass through one interior point of the $25$-gon. (A Shapovalov)

1997 Czech And Slovak Olympiad IIIA, 6

Tags: cyclic , parallelogram , locus , diagonal , geometry

In a parallelogram $ABCD$, triangle $ABD$ is acute-angled and $\angle BAD = \pi /4$. Consider all possible choices of points $K,L,M,N$ on sides $AB,BC, CD,DA$ respectively, such that $KLMN$ is a cyclic quadrilateral whose circumradius equals those of triangles $ANK$ and $CLM$. Find the locus of the intersection of the diagonals of $KLMN$

2017 Czech-Polish-Slovak Junior Match, 2

Tags: hexagon , geometry , diagonal

Decide if exists a convex hexagon with all sides longer than $1$ and all nine of its diagonals are less than $2$ in length.

1998 Czech And Slovak Olympiad IIIA, 5

Tags: geometry , trapezoid , fixed point , diagonal

A circle $k$ and a point $A$ outside it are given in the plane. Prove that all trapezoids, whose non-parallel sides meet at $A$, have the same intersection of diagonals.

2014 Hanoi Open Mathematics Competitions, 2

Tags: combinatorial geometry , geometry , convex polygon , diagonal

How many diagonals does $11$-sided convex polygon have?

1991 Austrian-Polish Competition, 6

Tags: bisects , geometry , area of a triangle , diagonal , convex , area

Suppose that there is a point $P$ inside a convex quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal areas. Prove that one of the diagonals bisects the area of $ABCD$.

2020 Nordic, 3

Tags: trisector , sum , area of a triangle , geometry , diagonal , area

Each of the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ is divided into three equal parts, $|AE| = |EF| = |F B|$ , $|DP| = |P Q| = |QC|$. The diagonals of $AEPD$ and $FBCQ$ intersect at $M$ and $N$, respectively. Prove that the sum of the areas of $\vartriangle AMD$ and $\vartriangle BNC$ is equal to the sum of the areas of $\vartriangle EPM$ and $\vartriangle FNQ$.

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 $1$?

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.

2018 Oral Moscow Geometry Olympiad, 1

Tags: parallelogram , diagonal , geometry

Two parallelograms are arranged so as it shown on the picture. Prove that the diagonal of the one parallelogram passes through the intersection point of the diagonals of the second. [img]https://cdn.artofproblemsolving.com/attachments/9/a/15c2f33ee70eec1bcc44f94ec0e809c9e837ff.png[/img]

Swiss NMO - geometry, 2008.8

Tags: geometry , hexagon , cyclic , concurrency , diagonal

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

2018 Oral Moscow Geometry Olympiad, 2

Tags: geometry , trapezoid , perpendicular , right angle , diagonal

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)