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: 34

1988 ITAMO, 3
Tags: radius , geometry , distance , regular , pentagon
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.

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?

Ukraine Correspondence MO - geometry, 2019.8
Tags: ratio , regular , pentagon , geometry
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, 2005.2.3
Tags: geometry , regular , octagon , 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.

I Soros Olympiad 1994-95 (Rus + Ukr), 9.2
Tags: combinatorics , combinatorial geometry , regular , equilateral , game , strategy
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.

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: 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.

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?

Estonia Open Senior - geometry, 2020.1.5
Tags: geometry , regular , pentagon , concurrency
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.

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]

1989 Tournament Of Towns, (232) 6
Tags: equal area , geometry , hexagon , regular , combinatorial geometry , parallelogram
A regular hexagon is cut up into $N$ parallelograms of equal area. Prove that $N$ is divisible by three. (V. Prasolov, I. Sharygin, Moscow)

1994 Poland - Second Round, 3
Tags: plane , cyclic , hexagon , regular , geometry , 3d geometry
A plane passing through the center of a cube intersects the cube in a cyclic hexagon. Show that this hexagon is regular.

1990 Greece Junior Math Olympiad, 3
Tags: regular , angle , geometry
Let $A_1A_2A_3...A_{72}$ be a regurar $72$-gon with center $O$. Calculate an extenral angle of that polygon and the angles $\angle A_{45} OA_{46}$, $\angle A_{44} A_{45}A_{46}$. How many diagonals does this polygon have?

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.

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

1979 Chisinau City MO, 173
Tags: equal angles , pentagon , regular , geometry
The inner angles of the pentagon inscribed in the circle are equal to each other. Prove that this pentagon is regular.

2003 Estonia National Olympiad, 1
Tags: geometry , regular , pentagon , circles , area
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)

2006 Sharygin Geometry Olympiad, 4
Tags: square , geometry , regular , 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 $MS$.

2011 Denmark MO - Mohr Contest, 2
Tags: geometry , regular , octagon , distance
In the octagon below all sides have the length $1$ and all angles are equal. Determine the distance between the corners $A$ and $B$. [img]https://1.bp.blogspot.com/-i6TAFDvcQ8w/XzXCRhnV_kI/AAAAAAAAMVw/rKrQMfPYYJIaCwl8hhdVHdqO4fIn8O7cwCLcBGAsYHQ/s0/2011%2BMogh%2Bp2.png[/img]

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]

2007 Abels Math Contest (Norwegian MO) Final, 2
Tags: ratio , cyclic , pentagon , circumradius , circles , regular , geometry
The vertices of a convex pentagon $ABCDE$ lie on a circle $\gamma_1$. The diagonals $AC , CE, EB, BD$, and $DA$ are tangents to another circle $\gamma_2$ with the same centre as $\gamma_1$. (a) Show that all angles of the pentagon $ABCDE$ have the same size and that all edges of the pentagon have the same length. (b) What is the ratio of the radii of the circles $\gamma_1$ and $\gamma_2$? (The answer should be given in terms of integers, the four basic arithmetic operations and extraction of roots only.)

Denmark (Mohr) - geometry, 2011.2
Tags: geometry , regular , octagon , distance
In the octagon below all sides have the length $1$ and all angles are equal. Determine the distance between the corners $A$ and $B$. [img]https://1.bp.blogspot.com/-i6TAFDvcQ8w/XzXCRhnV_kI/AAAAAAAAMVw/rKrQMfPYYJIaCwl8hhdVHdqO4fIn8O7cwCLcBGAsYHQ/s0/2011%2BMogh%2Bp2.png[/img]

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?

Estonia Open Junior - geometry, 2020.1.5
Tags: geometry , perpendicular , regular , pentagon
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.