AoPS - Problem

Processing math: 93%

Problem

Source: Cono Sur Olympiad 2017, problem 2

Tags: geometry, cono sur

adrian97

18.08.2017 02:40

Let $A(XYZ)$ be the area of the triangle $XYZ$ . A non-regular convex polygon $P_1 P_2 \ldots P_n$ is called guayaco if exists a point $O$ in its interior such that $A(P_1OP_2) = A(P_2OP_3) = \cdots = A(P_nOP_1).$ Show that, for every integer $n \ge 3$ , a guayaco polygon of $n$ sides exists.

62861

18.08.2017 03:05

Take a regular $n$ -gon and apply a non-trivial affine transformation to it. Done!

duck_master

18.08.2017 04:11

CantonMathGuy wrote: Take a regular $n$ -gon and apply a non-trivial affine transformation to it. Done! And every affine transformation scales area, right?

brainiac1

18.08.2017 04:56

Affine transformations preserve ratios of areas.

mathisreal

29.08.2017 22:39

How did you solve this problem?

LaPlanta

30.08.2017 21:17

If $n=3$ , choose any non equilateral triangle and let $O$ be the barycenter. If $n=4$ , choose a paralelogram that isn't a square and let $O$ be the intersection of the diagonals. For $n\geq5$ , take $n-2$ congruent isosceles triangles with an angle of $\frac{180º}{n-2}$ in such a way that they add up to a flat angle. Then add 2 rectangle triangles with matching areas at the bottom to finish the polygon: