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.

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Found problems: 179

2011 Saudi Arabia Pre-TST, 4.2
Tags: pentagon , cyclic , geometry
Pentagon $ABCDE$ is inscribed in a circle. Distances from point $E$ to lines $AB$ , $BC$ and $CD$ are equal to $a, b$ and $c$, respectively. Find the distance from point $E$ to line $AD$.

II Soros Olympiad 1995 - 96 (Russia), 9.3
Tags: geometry , combinatorial geometry , pentagon
Is there a convex pentagon in which each diagonal is equal to some side?

2021 Canadian Mathematical Olympiad Qualification, 3
Tags: geometry , pentagon
$ABCDE$ is a regular pentagon. Two circles $C_1$ and $C_2$ are drawn through $B$ with centers $A$ and $C$ respectively. Let the other intersection of $C_1$ and $C_2$ be $P$. The circle with center $P$ which passes through $E$ and $D$ intersects $C_2$ at $X$ and $AE$ at $Y$. Prove that $AX = AY$.

2018 Germany Team Selection Test, 2
Tags: geometry , pentagon , inscribed circle
Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2004 Estonia Team Selection Test, 6
Tags: pentagon , hexagon , combinatorics , combinatorial geometry , 3d geometry , geometry , polyhedron
Call a convex polyhedron a [i]footballoid [/i] if it has the following properties. (1) Any face is either a regular pentagon or a regular hexagon. (2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it). Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.

Champions Tournament Seniors - geometry, 2001.4
Tags: pentagon , equal angles , equal segments , geometry
Given a convex pentagon $ABCDE$ in which $\angle ABC = \angle AED = 90^o$, $\angle BAC= \angle DAE$. Let $K$ be the midpoint of the side $CD$, and $P$ the intersection point of lines $AD$ and $BK$, $Q$ be the intersection point of lines $AC$ and $EK$. Prove that $BQ = PE$.

2009 Brazil Team Selection Test, 1
Tags: right triangle , orthocenter , geometry , cyclic , pentagon
Let $A, B, C, D, E$ points in circle of radius r, in that order, such that $AC = BD = CE = r$. The points $H_1, H_2, H_3$ are the orthocenters of the triangles $ACD$, $BCD$ and $BCE$, respectively. Prove that $H_1H_2H_3$ is a right triangle .

Estonia Open Junior - geometry, 1996.2.4
Tags: geometry , trigonometry , pentagon , area
A pentagon (not necessarily convex) has all sides of length $1$ and its product of cosine of any four angles equal to zero. Find all possible values of the area of such a pentagon.

2016 NIMO Problems, 3
Tags: geometry , pentagon
Convex pentagon $ABCDE$ satisfies $AB \parallel DE$, $BE \parallel CD$, $BC \parallel AE$, $AB = 30$, $BC = 18$, $CD = 17$, and $DE = 20$. Find its area. [i] Proposed by Michael Tang [/i]

2007 Germany Team Selection Test, 2
Tags: circumcircle , pentagon , geometry
Let $ ABCDE$ be a convex pentagon such that \[ \angle BAC \equal{} \angle CAD \equal{} \angle DAE\qquad \text{and}\qquad \angle ABC \equal{} \angle ACD \equal{} \angle ADE. \]The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$. [i]Proposed by Zuming Feng, USA[/i]

2012 Korea Junior Math Olympiad, 2
Tags: geometry , pentagon , concyclic
A pentagon $ABCDE$ is inscribed in a circle $O$, and satis fies $\angle A = 90^o, AB = CD$. Let $F$ be a point on segment $AE$. Let $BF$ hit $O$ again at $J(\ne B)$, $CE \cap DJ = K$, $BD\cap FK = L$. Prove that $B,L,E,F$ are cyclic.

May Olympiad L1 - geometry, 2007.5
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.)

1966 Spain Mathematical Olympiad, 3
Tags: geometry , similarity , pentagon
Given a regular pentagon, consider the convex pentagon limited by its diagonals. You are asked to calculate: a) The similarity relation between the two convex pentagons. b) The relationship of their areas. c) The ratio of the homothety that transforms the first into the second.

2008 Korea Junior Math Olympiad, 5
Tags: tangent , geometry , pentagon , concurrency
Let there be a pentagon $ABCDE$ inscribed in a circle $O$. The tangent to $O$ at $E$ is parallel to $AD$. A point $F$ lies on $O$ and it is in the opposite side of $A$ with respect to $CD$, and satisfi es $AB \cdot BC \cdot DF = AE \cdot ED \cdot CF$ and $\angle CFD = 2\angle BFE$. Prove that the tangent to $O$ at $B,E$ and line $AF$ concur at one point.

2021 Yasinsky Geometry Olympiad, 1
Tags: geometry , perimeter , pentagon , dodecagon
A regular dodecagon $A_1A_2...A_{12}$ is inscribed in a circle with a diameter of $20$ cm . Calculate the perimeter of the pentagon $A_1A_3A_6A_8A_{11}$. (Alexey Panasenko)

2021 Polish Junior MO First Round, 7
Tags: geometry , pentagon , folding , paper folding
The figure below, composed of four regular pentagons with a side length of $1$, was glued in space as follows. First, it was folded along the broken sections, by combining the bold sections, and then formed in such a way that colored sections formed a square. Find the length of the segment $AB$ created in this way. [img]https://cdn.artofproblemsolving.com/attachments/0/7/bddad6449f74cbc7ea2623957ef05b3b0d2f19.png[/img]

2018 Brazil Team Selection Test, 1
Tags: geometry , pentagon , inscribed circle
Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2018 Hanoi Open Mathematics Competitions, 7
Tags: pentagon , geometry
Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^o,\angle B = 105^o,\angle C = 90^o$ and $AB = 2,BC = CD = DE =\sqrt2$. If the length of $AE$ is $\sqrt{a }- b$ where $a, b$ are integers, what is the value of $a + b$?

2002 IMO Shortlist, 5
Tags: geometry , ratio , area , triangle , pentagon
For any set $S$ of five points in the plane, no three of which are collinear, let $M(S)$ and $m(S)$ denote the greatest and smallest areas, respectively, of triangles determined by three points from $S$. What is the minimum possible value of $M(S)/m(S)$ ?

1986 IMO, 3
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.

2016 Bulgaria JBMO TST, 2
Tags: geometry , square , orthocenter , cyclic , pentagon
The vertices of the pentagon $ABCDE$ are on a circle, and the points $H_1, H_2, H_3,H_4$ are the orthocenters of the triangles $ABC, ABE, ACD, ADE$ respectively . Prove that the quadrilateral determined by the four orthocenters is square if and only if $BE \parallel CD$ and the distance between them is $\frac{BE + CD}{2}$.

2018 Germany Team Selection Test, 2
Tags: geometry , pentagon , inscribed circle
Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2020 Ukrainian Geometry Olympiad - April, 5
Tags: geometry , parallelogram , pentagon , equal angles , equal segments
Given a convex pentagon $ABCDE$, with $\angle BAC = \angle ABE = \angle DEA - 90^o$, $\angle BCA = \angle ADE$ and also $BC = ED$. Prove that $BCDE$ is parallelogram.

2007 Alexandru Myller, 3
Tags: geometry , pentagon
The convex pentagon $ ABCDE $ has the following properties: $ \text{(i)} AB=BC $ $ \text{(ii)} \angle ABE+\angle CBD =\angle DBE $ $ \text{(iii)} \angle AEB +\angle BDC=180^{\circ} $ Prove that the orthocenter of $ BDE $ lies on $ AC. $

2000 Tuymaada Olympiad, 7
Tags: combinatorics , combinatorial geometry , geometry , regular polygon , pentagon
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?