Olympiad problems

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]

1989 Tournament Of Towns, (232) 6

Tags: regular , equal area , hexagon , geometry , parallelogram , combinatorial geometry

A regular hexagon is cut up into $N$ parallelograms of equal area. Prove that $N$ is divisible by three. (V. Prasolov, I. Sharygin, Moscow)

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.

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.

1904 Eotvos Mathematical Competition, 1

Tags: regular , pentagon , geometry

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

2006 Sharygin Geometry Olympiad, 4

Tags: geometry , regular , square , pentagon

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 $MS$.

Estonia Open Senior - geometry, 2020.1.5

Tags: regular , pentagon , concurrency , 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 $F$. 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.

Estonia Open Junior - geometry, 2005.2.3

Tags: regular , octagon , geometry , square

The vertices of the square $ABCD$ are the centers of four circles, all of which pass through the center of the square. Prove that the intersections of the circles on the square $ABCD$ sides are vertices of a regular octagon.

2012 NZMOC Camp Selection Problems, 1

Tags: regular , octagon , area , square , geometry

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?

2004 Junior Balkan Team Selection Tests - Romania, 4

Tags: coloring , regular , combinatorics

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

I Soros Olympiad 1994-95 (Rus + Ukr), 9.2

Tags: strategy , regular , equilateral , combinatorics , combinatorial geometry , game

Given a regular $72$-gon. Lenya and Kostya play the game "Make an equilateral triangle." They take turns marking with a pencil on one still unmarked angle of the $72$-gon: Lenya uses red. Kostya uses blue. Lenya starts the game, and the one who marks first wins if its color is three vertices that are the vertices of some equilateral triangle, if all the vertices are marked and no such a triangle exists, the game ends in a draw. Prove that Kostya can play like this so as not to lose.

2019 New Zealand MO, 6

Tags: regular , combinatorics

Let $V$ be the set of vertices of a regular $21$-gon. Given a non-empty subset $U$ of $V$ , let $m(U)$ be the number of distinct lengths that occur between two distinct vertices in $U$. What is the maximum value of $\frac{m(U)}{|U|}$ as $U$ varies over all non-empty subsets of $V$ ?

Ukrainian TYM Qualifying - geometry, 2010.16

Tags: 3d geometry , regular , sphere , tetrahedron , geometry

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 Junior - geometry, 2011.2.3

Tags: hexagon , regular , geometry

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?

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 $AB =\frac{mn}{m+n}$.

Estonia Open Senior - geometry, 2019.1.5

Tags: polygon , regular , equal angles , geometry

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 , regular , equal segments

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]

1994 Poland - Second Round, 3

Tags: hexagon , geometry , plane , cyclic , regular , 3d geometry

A plane passing through the center of a cube intersects the cube in a cyclic hexagon. Show that this hexagon is regular.

1991 Romania Team Selection Test, 9

Tags: similar , regular , pentagon , geometry

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

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]

1997 Tournament Of Towns, (563) 4

Tags: combinatorial geometry , geometry , regular , combinatorics

(a) Several identical napkins, each in the shape of a regular hexagon, are put on a table (the napkins may overlap). Each napkin has one side which is parallel to a fixed line. Is it always possible to hammer a few nails into the table so that each napkin is nailed with exactly one nail? (b) The same question for regular pentagons. (A Kanel)

1988 ITAMO, 3

Tags: radius , distance , regular , pentagon , geometry

A regular pentagon of side length $1$ is given. Determine the smallest $r$ for which the pentagon can be covered by five discs of radius $r$ and justify your answer.

Ukraine Correspondence MO - geometry, 2019.8

Tags: geometry , regular , pentagon , ratio

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]

Denmark (Mohr) - geometry, 2023.4

Tags: equal segments , regular , geometry

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]

1979 Chisinau City MO, 173

Tags: equal angles , pentagon , geometry , regular

The inner angles of the pentagon inscribed in the circle are equal to each other. Prove that this pentagon is regular.