Found problems: 7
Novosibirsk Oral Geo Oly VII, 2020.1
All twelve points on the circle are at equal distances. The only marked point inside is the center of the circle. Determine which part of the whole circle in the picture is filled in.
[img]https://cdn.artofproblemsolving.com/attachments/9/0/9a6af9cef6a4bb03fb4d3eef715f3fd77c74b3.png[/img]
Brazil L2 Finals (OBM) - geometry, 2006.2
Among the $5$-sided polygons, as many vertices as possible collinear , that is, belonging to a single line, is three, as shown below. What is the largest number of collinear vertices a $12$-sided polygon can have?
[img]https://cdn.artofproblemsolving.com/attachments/1/1/53d419efa4fc4110730a857ae6988fc923eb13.png[/img]
Attention: In addition to drawing a $12$-sided polygon with the maximum number of vertices collinear , remember to show that there is no other $12$-sided polygon with more vertices collinear than this one.
1971 All Soviet Union Mathematical Olympiad, 154
a) The vertex $A_1$ of the regular $12$-gon (dodecagon) $A_1A_2...A_{12}$ is marked with "$-$" and all the rest $--$ with "$+$". You are allowed to change the sign simultaneously in the $6$ vertices in succession. Prove that is impossible to obtain dodecagon with $A_2$ marked with "$-$" and the rest of the vertices $--$ with "$+$".
b) Prove the same statement if it is allowed to change the signs not in six, but in four vertices in succession.
c) Prove the same statement if it is allowed to change the signs in three vertices in succession.
1986 All Soviet Union Mathematical Olympiad, 431
Given two points inside a convex dodecagon (twelve sides) situated $10$ cm far from each other. Prove that the difference between the sum of distances, from the point to all the vertices, is less than $1$ m for those points.
2024 Mexico National Olympiad, 3
Let $ABCDEF$ a convex hexagon, and let $A_1,B_1,C_1,D_1,E_1$ y $F_1$ be the midpoints of $AB,BC,CD,$ $DE,EF$ and $FA$, respectively. Construct points $A_2,B_2,C_2,D_2,E_2$ and $F_2$ in the interior of $A_1B_1C_1D_1E_1F_1$ such that both
1. The sides of the dodecagon $A_2A_1B_2B_1C_2C_1D_2D_1E_2E_1F_2F_1$ are all equal and
2. $\angle A_1B_2B_1+\angle C_1D_2D_1+\angle E_1F_2F_1=\angle B_1C_2C_1+\angle D_1E_2E_1+\angle F_1A_2A_1=360^\circ$, where all these angles are less than $180 ^\circ$,
Prove that $A_2B_2C_2D_2E_2F_2$ is cyclic.
[b]Note:[/b] Dodecagon $A_2A_1B_2B_1C_2C_1D_2D_1E_2E_1F_2F_1$ is shaped like a 6-pointed star, where the points are $A_1,B_1,C_1,D_1,E_1$ y $F_1$.
2021 Yasinsky Geometry Olympiad, 1
A regular dodecagon $A_1A_2...A_{12}$ is inscribed in a circle with a diameter of $20$ cm . Calculate the perimeter of the pentagon $A_1A_3A_6A_8A_{11}$.
(Alexey Panasenko)
2020 Novosibirsk Oral Olympiad in Geometry, 1
All twelve points on the circle are at equal distances. The only marked point inside is the center of the circle. Determine which part of the whole circle in the picture is filled in.
[img]https://cdn.artofproblemsolving.com/attachments/9/0/9a6af9cef6a4bb03fb4d3eef715f3fd77c74b3.png[/img]