Olympiad problems

1991 Romania Team Selection Test, 9

The diagonals of a pentagon $ABCDE$ determine another pentagon $MNPQR$. If $MNPQR$ and $ABCDE$ are similar, must $ABCDE$ be regular?

Champions Tournament Seniors - geometry, 2001.4

Tags: equal segments , pentagon , equal angles , 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 $BQ = PE$.

Durer Math Competition CD 1st Round - geometry, 2023.C7

Tags: geometry , angle , pentagon

Let $ABCDE$ be a regular pentagon. We drew two circles around $A$ and $B$ with radius $AB$. Let $F$ mark the intersection of the two circles that is inside the pentagon. Let $G$ mark the intersection of lines $EF$ and $AD$. What is the degree measure of angle $AGE$?

2013 Sharygin Geometry Olympiad, 1

Tags: parallel , cyclic , pentagon , geometry

All angles of a cyclic pentagon $ABCDE$ are obtuse. The sidelines $AB$ and $CD$ meet at point $E_1$, the sidelines $BC$ and $DE$ meet at point $A_1$. The tangent at $B$ to the circumcircle of the triangle $BE_1C$ meets the circumcircle $\omega$ of the pentagon for the second time at point $B_1$. The tangent at $D$ to the circumcircle of the triangle $DA_1C$ meets $\omega$ for the second time at point $D_1$. Prove that $B_1D_1 // AE$

1981 IMO Shortlist, 14

Tags: geometry , pentagon , circumcircle

Prove that a convex pentagon (a five-sided polygon) $ABCDE$ with equal sides and for which the interior angles satisfy the condition $\angle A \geq \angle B \geq \angle C \geq \angle D \geq \angle E$ is a regular pentagon.

2021 Sharygin Geometry Olympiad, 9.2

Tags: geometry , geometric inequality , pentagon , ratio

A cyclic pentagon is given. Prove that the ratio of its area to the sum of the diagonals is not greater than the quarter of the circumradius.

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]

1997 German National Olympiad, 6b

Tags: pentagon , combinatorics , combinatorial geometry

An approximate construction of a regular pentagon goes as follows. Inscribe an arbitrary convex pentagon $P_1P_2P_3P_4P_5$ in a circle. Now choose an arror bound $\epsilon > 0$ and apply the following procedure. (a) Denote $P_0 = P_5$ and $P_6 = P_1$ and construct the midpoint $Q_i$ of the circular arc $P_{i-1}P_{i+1}$ containing $P_i$. (b) Rename the vertices $Q_1,...,Q_5$ as $P_1,...,P_5$. (c) Repeat this procedure until the difference between the lengths of the longest and the shortest among the arcs $P_iP_{i+1}$ is less than $\epsilon$. Prove this procedure must end in a finite time for any choice of $\epsilon$ and the points $P_i$.

2013 Hanoi Open Mathematics Competitions, 8

Tags: area , geometry , pentagon

Let $ABCDE$ be a convex pentagon and area of $\vartriangle ABC =$ area of $\vartriangle BCD =$ area of $\vartriangle CDE=$ area of $\vartriangle DEA =$ area of $\vartriangle EAB$. Given that area of $\vartriangle ABCDE = 2$. Evaluate the area of area of $\vartriangle ABC$.

2014 Saudi Arabia GMO TST, 3

Tags: cyclic , pentagon , geometry

Let $ABCDE$ be a cyclic pentagon such that the diagonals $AC$ and $AD$ intersect $BE$ at $P$ and $Q$, respectively, with $BP \cdot QE = PQ^2$. Prove that $BC \cdot DE = CD \cdot PQ$.

1988 IMO Shortlist, 17

Tags: convex polygon , angle , pentagon , geometry

In the convex pentagon $ ABCDE,$ the sides $ BC, CD, DE$ are equal. Moreover each diagonal of the pentagon is parallel to a side ($ AC$ is parallel to $ DE$, $ BD$ is parallel to $ AE$ etc.). Prove that $ ABCDE$ is a regular pentagon.

2020 Malaysia IMONST 1, 12

Tags: hexagon , pentagon

A football is made by sewing together some black and white leather patches. The black patches are regular pentagons of the same size. The white patches are regular hexagons of the same size. Each pentagon is bordered by 5 hexagons. Each hexagons is bordered by $3$ pentagons and $3$ hexagons. We need $12$ pentagons to make one football. How many hexagons are needed to make one football?

2012 China Northern MO, 7

Tags: bisects segment , pentagon , geometry

As shown in figure , in the pentagon $ABCDE$, $BC = DE$, $CD \parallel BE$, $AB>AE$. If $\angle BAC = \angle DAE$ and $\frac{AB}{BD}=\frac{AE}{ED}$. Prove that $AC$ bisects the line segment $BE$. [img]https://cdn.artofproblemsolving.com/attachments/3/2/5ce44f1e091786b865ae4319bda56c3ddbb8d7.png[/img]

1987 All Soviet Union Mathematical Olympiad, 458

Tags: area , diagonal , convex , pentagon

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.

JBMO Geometry Collection, 2009

Tags: trapezoid , pentagon , geometry

Let $ ABCDE$ be a convex pentagon such that $ AB\plus{}CD\equal{}BC\plus{}DE$ and $ k$ a circle with center on side $ AE$ that touches the sides $ AB$, $ BC$, $ CD$ and $ DE$ at points $ P$, $ Q$, $ R$ and $ S$ (different from vertices of the pentagon) respectively. Prove that lines $ PS$ and $ AE$ are parallel.

2020 Ukrainian Geometry Olympiad - April, 5

Tags: parallelogram , equal angles , pentagon , equal segments , geometry

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.

2018 NZMOC Camp Selection Problems, 6

Tags: geometry , cube , 3d geometry , pentagon

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.

2025 6th Memorial "Aleksandar Blazhevski-Cane", P4

Tags: geometry , circumcircle , concurrency , pentagon

Let $ABCDE$ be a pentagon such that $\angle DCB < 90^{\circ} < \angle EDC$. The circle with diameter $BD$ intersects the line $BC$ again at $F$, and the circle with diameter $DE$ intersects the line $CE$ again at $G$. Prove that the second intersection ($\neq D$) of the circumcircle of $\triangle DFG$ and the circle with diameter $AD$ lies on $AC$. Proposed by [i]Petar Filipovski[/i]

1979 Chisinau City MO, 182

Tags: cube , plane , 3d geometry , pentagon , geometry

Prove that a section of a cube by a plane cannot be a regular pentagon.

1997 Belarusian National Olympiad, 1

Tags: equal segments , cyclic , geometry , perpendicular , pentagon

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 Estonia Team Selection Test, 6

Tags: hexagon , 3d geometry , polyhedron , combinatorial geometry , pentagon , combinatorics , geometry

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.

1969 IMO Longlists, 50

Tags: angle bisector , pentagon , construction , geometry

$(NET 5)$ The bisectors of the exterior angles of a pentagon $B_1B_2B_3B_4B_5$ form another pentagon $A_1A_2A_3A_4A_5.$ Construct $B_1B_2B_3B_4B_5$ from the given pentagon $A_1A_2A_3A_4A_5.$

2016 Bulgaria JBMO TST, 2

Tags: orthocenter , cyclic , pentagon , geometry , square

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 $\frac{BE + CD}{2}$.

1995 Mexico National Olympiad, 5

Tags: area , geometry , pentagon

$ABCDE$ is a convex pentagon such that the triangles $ABC, BCD, CDE, DEA$ and $EAB$ have equal areas. Show that $(1/4)$ area $(ABCDE) <$ area $(ABC) < (1/3)$ area $(ABCDE)$.

Russian TST 2019, P1

Tags: geometry , pentagon

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 $\Gamma$.