Olympiad problems

1999 May Olympiad, 2

In a parallelogram $ABCD$ , $BD$ is the largest diagonal. By matching $B$ with $D$ by a bend, a regular pentagon is formed. Calculate the measures of the angles formed by the diagonal $BD$ with each of the sides of the parallelogram.

1994 Tournament Of Towns, (407) 5

Tags: similar , geometry , pentagon , combinatorial geometry

Does there exist a convex pentagon from which a similar pentagon can be cut off by a straight line? (S Tokarev)

2003 Singapore MO Open, 4

Tags: 3d geometry , geometric inequalities , geometry , pentagon , pyramid

The pentagon $ABCDE$ which is inscribed in a circle with $AB < DE$ is the base of a pyramid with apex $S$. If the longest side from $S$ is $SA$, prove that $BS > CS$.

2003 Estonia National Olympiad, 1

Tags: regular , circles , pentagon , area , geometry

The picture shows $10$ equal regular pentagons where each two neighbouring pentagons have a common side. The smaller circle is tangent to one side of each pentagon and the larger circle passes through the opposite vertices of these sides. Find the area of the larger circle if the area of the smaller circle is $1$. [img]https://cdn.artofproblemsolving.com/attachments/0/6/84fe98370868a5cf28d92d4b207ccb00e6eaa3.png[/img]

2011 Bundeswettbewerb Mathematik, 3

Tags: trisector , diagonal , convex , pentagon , geometry

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

1953 Moscow Mathematical Olympiad, 242

Tags: equal area , pentagon , area , geometry

Let $A$ be a vertex of a regular star-shaped pentagon, the angle at $A$ being less than $180^o$ and the broken line $AA_1BB_1CC_1DD_1EE_1$ being its contour. Lines $AB$ and $DE$ meet at $F$. Prove that polygon $ABB_1CC_1DED_1$ has the same area as the quadrilateral $AD_1EF$. Note: A regular star pentagon is a figure formed along the diagonals of a regular pentagon.

Novosibirsk Oral Geo Oly VII, 2023.4

Tags: congruent triangles , congruence , pentagon , geometry

Inside the convex pentagon $ABCDE$, a point $O$ was chosen, and it turned out that all five triangles $AOB$, $BOC$, $COD$, $DOE$ and $EOA$ are congrunet to each other. Prove that these triangles are isosceles or right-angled.

2025 JBMO TST - Turkey, 7

Tags: pentagon , ratio , geometry

$ABCDE$ is a pentagon whose vertices lie on circle $\omega$ where $\angle DAB=90^{\circ}$. Let $EB$ and $AC$ intersect at $F$, $EC$ meet $BD$ at $G$. $M$ is the midpoint of arc $AB$ on $\omega$, not containing $C$. If $FG\parallel DE\parallel CM$ holds, then what is the value of $\frac{|GE|}{|GD|}$?

1985 All Soviet Union Mathematical Olympiad, 404

Tags: midpoint , pentagon , geometric construction , geometry , symmetry

The convex pentagon $ABCDE$ was drawn in the plane. $A_1$ was symmetric to $A$ with respect to $B$. $B_1$ was symmetric to $B$ with respect to $C$. $C_1$ was symmetric to $C$ with respect to $D$. $D_1$ was symmetric to $D$ with respect to $E$. $E_1$ was symmetric to $E$ with respect to $A$. How is it possible to restore the initial pentagon with the compasses and ruler, knowing $A_1,B_1,C_1,D_1,E_1$ points?

2011 Belarus Team Selection Test, 1

Tags: pentagon , geometric inequality , geometry

Let $A$ be the sum of all $10$ distinct products of the sides of a convex pentagon, $S$ be the area of the pentagon. a) Prove thas $S \le \frac15 A$. b) Does there exist a constant $c<\frac15$ such that $S \le cA$ ? I.Voronovich

2012 Oral Moscow Geometry Olympiad, 2

Tags: pentagon , area , geometry

In the convex pentagon $ABCDE$: $\angle A = \angle C = 90^o$, $AB = AE, BC = CD, AC = 1$. Find the area of the pentagon.

1991 Tournament Of Towns, (286) 2

Tags: perpendicular , pentagon , geometry

The pentagon $ABCDE$ has an inscribed circle and the diagonals $AD$ and $CE$ intersect in its centre $O$. Prove that the segment $BO$ and the side $DE$ are perpendicular. (Folklore)

2008 Hanoi Open Mathematics Competitions, 7

Tags: convex , angle , pentagon , geometry

The figure $ABCDE$ is a convex pentagon. Find the sum $\angle DAC + \angle EBD +\angle ACE +\angle BDA + \angle CEB$?

2007 May Olympiad, 5

Tags: geometry , paper , rectangle , pentagon

You have a paper pentagon, $ABCDE$, such that $AB = BC = 3$ cm, $CD = DE= 5$ cm, $EA = 4$ cm, $\angle ABC = 100^o$ ,$ \angle CDE = 80^o$. You have to divide the pentagon into four triangles, by three straight cuts, so that with the four triangles assemble a rectangle, without gaps or overlays. (The triangles can be rotated and / or turned around.)

2014 Oral Moscow Geometry Olympiad, 3

Tags: convex , pentagon , diagonal , geometry

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

1993 Mexico National Olympiad, 3

Tags: combinatorial geometry , geometry , pentagon , area of a triangle

Given a pentagon of area $1993$ and $995$ points inside the pentagon, let $S$ be the set containing the vertices of the pentagon and the $995$ points. Show that we can find three points of $S$ which form a triangle of area $\le 1$.

2003 IMAR Test, 1

Tags: disc , regular , covering , pentagon , geometry

Prove that the interior of a convex pentagon whose sides are all equal, is not covered by the open disks having the sides of the pentagon as diameter.

1998 All-Russian Olympiad Regional Round, 10.6

Tags: convex , pentagon , geometry

The pentagon $A_1A_2A_3A_4A_5$ contains bisectors $\ell_1$, $\ell_2$, $...$, $\ell_5$ of angles $\angle A_1$, $\angle A_2$, $ ...$ , $\angle A_5$ respectively. Bisectors $\ell_1$ and $\ell_2$ intersect at point $B_1$, $\ell_2$ and $\ell_3$ - at point $B_2$, etc., $\ell_5$ and $\ell_1$ intersect at point $B_5$. Can the pentagon $B_1B_2B_3B_4B_5$ be convex?

2005 Sharygin Geometry Olympiad, 23

Tags: geometric transformation , 3d geometry , equal , pentagon , geometry

Envelop the cube in one layer with five convex pentagons of equal areas.

Estonia Open Junior - geometry, 2020.1.5

Tags: perpendicular , regular , pentagon , geometry

A circle $c$ with center $A$ passes through the vertices $B$ and $E$ of a regular pentagon $ABCDE$. The line $BC$ intersects the circle $c$ for second time at point $F$. Prove that the lines $DE$ and $EF$ are perpendicular.

2001 Regional Competition For Advanced Students, 3

Tags: area of a triangle , convex , pentagon , geometry , area

In a convex pentagon $ABCDE$, the area of the triangles $ABC, ABD, ACD$ and $ADE$ are equal and have the value $F$. What is the area of the triangle $BCE$ ?

2022 Iranian Geometry Olympiad, 3

Tags: geometry , pentagon

Let $ABCDE$ be a convex pentagon such that $AB = BC = CD$ and $\angle BDE = \angle EAC = 30 ^{\circ}$. Find the possible values of $\angle BEC$. [i]Proposed by Josef Tkadlec (Czech Republic)[/i]

2020 Canadian Mathematical Olympiad Qualification, 6

Tags: pentagon , geometry , orthocenter , concurrency

In convex pentagon $ABCDE, AC$ is parallel to $DE, AB$ is perpendicular to $AE$, and $BC$ is perpendicular to $CD$. If $H$ is the orthocentre of triangle $ABC$ and $M$ is the midpoint of segment $DE$, prove that $AD, CE$ and $HM$ are concurrent.

2000 Tuymaada Olympiad, 7

Tags: combinatorial geometry , geometry , pentagon , regular polygon , combinatorics

Every two of five regular pentagons on the plane have a common point. Is it true that some of these pentagons have a common point?

2015 Ukraine Team Selection Test, 9

Tags: point , heptagon , convex , combinatorial geometry , pentagon , combinatorics

The set $M$ consists of $n$ points on the plane and satisfies the conditions: $\bullet$ there are $7$ points in the set $M$, which are vertices of a convex heptagon, $\bullet$ for arbitrary five points with $M$, which are vertices of a convex pentagon, there is a point that also belongs to $M$ and lies inside this pentagon. Find the smallest possible value that $n$ can take .