4

Part of 2016 Sharygin Geometry Olympiad

Problems(3)

dissect a regular 10gon into 2regular polygons

Source: Sharygin Geometry Olympiad 2016 Final Round problem 4 grade 8

Is it possible to dissect a regular decagon along some of its diagonals so that the resulting parts can form two regular polygons?
by N.Beluhov

geometrypolygoncutregular polygon

Friendly Beetles crawling on a Plane

Source: Sharygin Geometry Olympiad, Final Round 2016, Problem 4 grade 9

One hundred and one beetles are crawling in the plane. Some of the beetles are friends. Every one hundred beetles can position themselves so that two of them are friends if and only if they are at unit distance from each other. Is it always true that all one hundred and one beetles can do the same?

combinatoricsgeometry

Betting money on sum of areas

Source: Sharygin geometry olympiad 2016, grade 10, Final Round, Problem 4.

The Devil and the Man play a game. Initially, the Man pays some cash

s

to the Devil. Then he lists some

97

\{i,j,k\}

consisting of positive integers not exceeding

100

. After that, the Devil draws some convex polygon

A_1A_2...A_{100}

100

and pays to the Man, the sum of areas of all triangles

A_iA_jA_k

. Determine the maximal value of

s

which guarantees that the Man receives at least as much cash as he paid.
Proposed by Nikolai Beluhov, Bulgaria

combinatoricsgeometryareagamegame strategy