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.
Problem
Source: Cono Sur Olympiad 2017, problem 2
Tags: geometry, cono sur
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 $\frac{180ยบ}{n-2}$ 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: