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

Ukrainian TYM Qualifying - geometry, 2010.16
Tags: geometry , 3d geometry , tetrahedron , sphere , regular
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.

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]

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.)

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.

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?

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?

2011 Denmark MO - Mohr Contest, 2
Tags: regular , octagon , distance , geometry
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]

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.

2019 New Zealand MO, 6
Tags: combinatorics , regular
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 $V$ ?

1989 Tournament Of Towns, (232) 6
Tags: hexagon , regular , combinatorial geometry , parallelogram , equal area , 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)

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 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.

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.

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]

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.

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 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?

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.

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

1997 Tournament Of Towns, (563) 4
Tags: geometry , combinatorial geometry , combinatorics , regular
(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)

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?

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

1990 Greece Junior Math Olympiad, 3
Tags: geometry , regular , angle
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?

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]