Olympiad problems

Denmark (Mohr) - geometry, 2011.2

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]

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)

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 $V$ ?

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.)

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.

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.

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]

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.

1989 Tournament Of Towns, (232) 6

Tags: geometry , hexagon , Regular , equal areas , 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)