Olympiad problems

2010 Saudi Arabia Pre-TST, 3.3

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

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]

2006 Sharygin Geometry Olympiad, 4

Tags: geometry , regular , pentagon , square

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, 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.

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.

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)

2010 Saudi Arabia Pre-TST, 3.3

1998 Portugal MO, 2

2006 Sharygin Geometry Olympiad, 4

Estonia Open Senior - geometry, 2019.1.5

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

2012 Belarus Team Selection Test, 1

Denmark (Mohr) - geometry, 2023.4

Estonia Open Junior - geometry, 2005.2.3

Kyiv City MO Juniors 2003+ geometry, 2005.89.5