Olympiad problems

Denmark (Mohr) - geometry, 2023.4

In the $9$-gon $ABCDEFGHI$, all sides have equal lengths and all angles are equal. Prove that $|AB| + |AC| = |AE|$. [img]https://cdn.artofproblemsolving.com/attachments/6/2/8c82e8a87bf8a557baaf6ac72b3d18d2ba3965.png[/img]

Kyiv City MO Juniors 2003+ geometry, 2005.89.5

Tags: geometry , angles , hexagon , Regular

Let $ABCDEF $ be a regular hexagon. On the line $AF $ mark the point $X$so that $ \angle DCX = 45^o$ . Find the value of the angle $FXE$. (Vyacheslav Yasinsky)

Estonia Open Senior - geometry, 2019.1.5

Tags: geometry , Regular , polygon , equal angles

Polygon $A_0A_1...A_{n-1}$ satisfies the following: $\bullet$ $A_0A_1 \le A_1A_2 \le ...\le A_{n-1}A_0$ and $\bullet$ $\angle A_0A_1A_2 = \angle A_1A_2A_3 = ... = \angle A_{n-2}A_{n-1}A_0$ (all angles are internal angles). Prove that this polygon is regular.

2023 Denmark MO - Mohr Contest, 4

Tags: geometry , equal segments , Regular

In the $9$-gon $ABCDEFGHI$, all sides have equal lengths and all angles are equal. Prove that $|AB| + |AC| = |AE|$. [img]https://cdn.artofproblemsolving.com/attachments/6/2/8c82e8a87bf8a557baaf6ac72b3d18d2ba3965.png[/img]

1991 Romania Team Selection Test, 9

Tags: similar , pentagon , geometry , Regular

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

2006 Sharygin Geometry Olympiad, 4

Tags: geometry , Regular , pentagon , Squares

a) Given two squares $ABCD$ and $DEFG$, with point $E$ lying on the segment $CD$, and points$ F,G$ outside the square $ABCD$. Find the angle between lines $AE$ and $BF$. b) Two regular pentagons $OKLMN$ and $OPRST$ are given, and the point $P$ lies on the segment $ON$, and the points $R, S, T$ are outside the pentagon $OKLMN$. Find the angle between straight lines $KP$ and $MS$.

Ukrainian TYM Qualifying - geometry, 2010.16

Tags: geometry , 3D geometry , tetrahedron , sphere , Regular , Ukrainian TYM

Points $A, B, C, D$ lie on the sphere of radius $1$. It is known that $AB\cdot AC\cdot AD\cdot BC\cdot BD\cdot CD=\frac{512}{27}$. Prove that $ABCD$ is a regular tetrahedron.

Estonia Open Senior - geometry, 2020.1.5

Tags: geometry , Regular , pentagon , concurrency , concurrent

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 $F$. The point $G$ on the circle $c$ is chosen such that $| F B | = | FG |$ and $B \ne G$. Prove that the lines $AB, EF$ and $DG$ intersect at one point.

1904 Eotvos Mathematical Competition, 1

Tags: geometry , Regular , pentagon

Prove that, if a pentagon (five-sided polygon) inscribed in a circle has equal angles, then its sides are equal.

2012 NZMOC Camp Selection Problems, 1

Tags: geometry , Regular , octagon , square , area

From a square of side length $1$, four identical triangles are removed, one at each corner, leaving a regular octagon. What is the area of the octagon?

1998 Portugal MO, 2

Tags: geometry , Regular , octagon

The regular octagon of the following figure is inscribed in a circle of radius $1$ and $P$ is a arbitrary point of this circle. Calculate the value of $PA^2 + PB^2 +...+ PH^2$. [img]https://cdn.artofproblemsolving.com/attachments/4/c/85e8e48c45970556077ac09c843193959b0e5a.png[/img]

2013 Portugal MO, 6

Tags: combinatorics , Regular , equal angles , combinatorial geometry , geometry

In each side of a regular polygon with $n$ sides, we choose a point different from the vertices and we obtain a new polygon of $n$ sides. For which values of $n$ can we obtain a polygon such that the internal angles are all equal but the polygon isn't regular?

2010 Saudi Arabia Pre-TST, 3.3

Tags: geometry , Heptagon , Regular

Let $ABCDEFG$ be a regular heptagon. If $AC = m$ and $AD = n$, prove that $AB =\frac{mn}{m+n}$.

2004 Junior Balkan Team Selection Tests - Romania, 4

Tags: Coloring , combinatorics , Regular

A regular polygon with $1000$ sides has the vertices colored in red, yellow or blue. A move consists in choosing to adjiacent vertices colored differently and coloring them in the third color. Prove that there is a sequence of moves after which all the vertices of the polygon will have the same color. Marius Ghergu

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.

2017 Romania National Olympiad, 1

Tags: geometry , 3D geometry , prism , Regular

Prove the following: a) If $ABCA'B'C'$ is a right prism and $M \in (BC), N \in (CA), P \in (AB)$ such that $A'M, B'N$ and $C'P$ are perpendicular each other and concurrent, then the prism $ABCA'B'C'$ is regular. b) If $ABCA'B'C'$ is a regular prism and $\frac{AA'}{AB}=\frac{\sqrt6}{4}$ , then there are $M \in (BC), N \in (CA), P \in (AB)$ so that the lines $A'M, B'N$ and $C'P$ are perpendicular each other and concurrent.

2015 Abels Math Contest (Norwegian MO) Final, 3

Tags: Regular , pentagon , distance , maximum , Product , geometry

The five sides of a regular pentagon are extended to lines $\ell_1, \ell_2, \ell_3, \ell_4$, and $\ell_5$. Denote by $d_i$ the distance from a point $P$ to $\ell_i$. For which point(s) in the interior of the pentagon is the product $d_1d_2d_3d_4d_5$ maximal?

2003 Estonia National Olympiad, 1

Tags: geometry , areas , Regular , pentagon , circles

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]

2012 Belarus Team Selection Test, 1

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

Let $m,n,k$ be pairwise relatively prime positive integers greater than $3$. Find the minimal possible number of points on the plane with the following property: there are $x$ of them which are the vertices of a regular $x$-gon for $x = m, x = n, x = k$. (E.Piryutko)

Ukraine Correspondence MO - geometry, 2019.8

Tags: ratio , Regular , pentagon , geometry , Ukraine Correspondence

The symbol of the Olympiad shows $5$ regular hexagons with side $a$, located inside a regular hexagon with side $b$. Find ratio $\frac{a}{b}$. [img]https://1.bp.blogspot.com/-OwyAl75LwiM/YIsThl3SG6I/AAAAAAAANS0/LwHEsAfyZMcqVIS8h_jr_n46OcMJaSTgQCLcBGAsYHQ/s0/2019%2BUkraine%2Bcorrespondence%2B5-12%2Bp8.png[/img]

Estonia Open Junior - geometry, 2011.2.3

Tags: geometry , hexagon , Regular

Consider the diagonals $A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_4$ and $A_6A_2$ of a convex hexagon $A_1A_2A_3A_4A_5A_6$. The hexagon whose vertices are the points of intersection of the diagonals is regular. Can we conclude that the hexagon $A_1A_2A_3A_4A_5A_6$ is also regular?