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.

AND:
OR:
NO:

Found problems: 179

1969 IMO Longlists, 70
Tags: geometry , geometric inequality , pentagon , euclidean distance
$(YUG 2)$ A park has the shape of a convex pentagon of area $50000\sqrt{3} m^2$. A man standing at an interior point $O$ of the park notices that he stands at a distance of at most $200 m$ from each vertex of the pentagon. Prove that he stands at a distance of at least $100 m$ from each side of the pentagon.

2001 Regional Competition For Advanced Students, 3
Tags: convex , pentagon , area of a triangle , area , geometry
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 $BCE$ ?

Ukrainian From Tasks to Tasks - geometry, 2010.13
Tags: geometry , pentagon , right angle , tangential
You can inscribe a circle in the pentagon $ABCDE$. It is also known that $\angle ABC = \angle BAE = \angle CDE = 90^o$. Find the measure of the angle $ADB$.

1986 IMO Shortlist, 12
Tags: invariant , algorithm , combinatorics , pentagon , game strategy
To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

Russian TST 2019, P1
Tags: pentagon , geometry
A convex pentagon $APBCQ$ is given such that $AB < AC$. The circle $\omega$ centered at point $A{}$ passes through $P{}$ and $Q{}$ and touches the segment $BC$ at point $R{}$. Let the circle $\Gamma$ centered at the point $O{}$ be the circumcircle of the triangle $ABC$. It is known that $AO \perp P Q$ and $\angle BQR = \angle CP R$. Prove that the tangents at points $P{}$ and $Q{}$ to the circle $\omega$ intersect on $\Gamma$.

2003 IMAR Test, 1
Tags: pentagon , covering , disc , regular , 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.

2022 Czech-Polish-Slovak Junior Match, 3
Tags: geometry , pentagon
Given is a convex pentagon $ABCDE$ in which $\angle A = 60^o$, $\angle B = 100^o$, $\angle C = 140^o$. Show that this pentagon can be placed in a circle with a radius of $\frac23 AD$.

2021 Flanders Math Olympiad, 2
Tags: geometry , pentagon
Catherine lowers five matching wooden discs over bars placed on the vertices of a regular pentagon. Then she leaves five smaller congruent checkers these rods drop. Then she stretches a ribbon around the large discs and a second ribbon around the small discs. The first ribbon has a length of $56$ centimeters and the second one of $50$ centimeters. Catherine looks at her construction from above and sees an area demarcated by the two ribbons. What is the area of that area? [img]https://cdn.artofproblemsolving.com/attachments/1/0/68e80530742f1f0775aff5a265e0c9928fa66c.png[/img]

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?

1969 All Soviet Union Mathematical Olympiad, 124
Tags: geometry , pentagon
Given a pentagon with all equal sides. a) Prove that there exist such a point on the maximal diagonal, that every side is seen from it inside a right angle. (side $AB$ is seen from the point $C$ inside an arbitrary angle that is greater or equal than $\angle ACB$) b) Prove that the circles constructed on its sides as on the diameters cannot cover the pentagon entirely.

2023 HMNT, 9
Tags: geometry , pentagon
Pentagon $SPEAK$ is inscribed in triangle $NOW$ such that $S$ and $P$ lie on segment $NO$, $K$ and $A$ lie on segment $NW$, and $E$ lies on segment $OW$. Suppose that $NS = SP = PO$ and $NK = KA = AW$. Given that $EP = EK = 5$ and $EA = ES = 6$, compute $OW$.

1997 Estonia National Olympiad, 3
Tags: parallel , pentagon , diagonal , ratio , geometry
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.

1967 Dutch Mathematical Olympiad, 3
Tags: parallel , pentagon , geometry
The convex pentagon $ABC DE$ is given, such that $AB,BC,CD$ and $DE$ are parallel to one of the diagonals. Prove that this also applies to $EA$.

2020 Canadian Mathematical Olympiad Qualification, 6
Tags: geometry , pentagon , 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.

1985 All Soviet Union Mathematical Olympiad, 415
Tags: pentagon , geometry , area
All the points situated more close than $1$ cm to ALL the vertices of the regular pentagon with $1$ cm side, are deleted from that pentagon. Find the area of the remained figure.

2018 NZMOC Camp Selection Problems, 6
Tags: pentagon , cube , 3d geometry , geometry
The intersection of a cube and a plane is a pentagon. Prove the length of at least oneside of the pentagon differs from 1 metre by at least 20 centimetres.

Brazil L2 Finals (OBM) - geometry, 2012.4
Tags: pentagon , angle chasing , region , equilateral , angle
The figure below shows a regular $ABCDE$ pentagon inscribed in an equilateral triangle $MNP$ . Determine the measure of the angle $CMD$. [img]http://4.bp.blogspot.com/-LLT7hB7QwiA/Xp9fXOsihLI/AAAAAAAAL14/5lPsjXeKfYwIr5DyRAKRy0TbrX_zx1xHQCK4BGAYYCw/s200/2012%2Bobm%2Bl2.png[/img]

1997 Moldova Team Selection Test, 2
Tags: ratio , pentagon , parallelogram , geometry
In a convex pentagon every diagonal is parallel to one side. Show that the ratios between the lengths of diagonals and the sides parallel to them are equal and find their value.

1991 Tournament Of Towns, (286) 2
Tags: geometry , pentagon , perpendicular
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)

2019 Thailand Mathematical Olympiad, 1
Tags: geometry , circumcircle , right triangle , pentagon , concyclic
Let $ABCDE$ be a convex pentagon with $\angle AEB=\angle BDC=90^o$ and line $AC$ bisects $\angle BAE$ and $\angle DCB$ internally. The circumcircle of $ABE$ intersects line $AC$ again at $P$. (a) Show that $P$ is the circumcenter of $BDE$. (b) Show that $A, C, D, E$ are concyclic.

1997 Belarusian National Olympiad, 1
Tags: geometry , equal segments , perpendicular , pentagon , cyclic
Different points $A_1,A_2,A_3,A_4,A_5$ lie on a circle so that $A_1A_2 = A_2A_3 = A_3A_4 =A_4A_5$. Let $A_6$ be the diametrically opposite point to $A_2$, and $A_7$ be the intersection of $A_1A_5$ and $A_3A_6$. Prove that the lines $A_1A_6$ and $A_4A_7$ are perpendicular

2004 Argentina National Olympiad, 5
Tags: geometry , area , pentagon , equal segments
The pentagon $ABCDE$ has $AB = BC, CD = DE, \angle ABC = 120^o, \angle CDE = 60^o$ and $BD = 2$. Calculate the area of the pentagon.

1979 Chisinau City MO, 182
Tags: geometry , cube , plane , pentagon , 3d geometry
Prove that a section of a cube by a plane cannot be a regular pentagon.

2003 Singapore MO Open, 4
Tags: 3d geometry , geometric inequalities , pentagon , pyramid , geometry
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 $BS > CS$.

2025 Kosovo National Mathematical Olympiad`, P1
Tags: geometry , easy , pentagon
The pentagon $ABCDE$ below is such that the quadrilateral $ABCD$ is a square and $BC=DE$. What is the measure of the angle $\angle AEC$?