 # Understand the problem

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.

Geometry
Easy
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Problem Solving Strategies by Arthur Engel.

Do you really need a hint? Try it first!

A regular polygon is obviously guayaco. As regular polygons are not allowed, we should try to “perturb” the polygon a little bit, preserving the areas.
Let us look at transformations of the plane of the form $(x,y)\mapsto (ax+by,cx+dy)$. Show that, if $|ad-bc|=1$ then this transformation preserves areas.
The class of transformations defined in hint 2 are known as area-preserving transformations. Clearly, taking the image of a regular $n$-gon under an area-preserving transformation should do the trick. However, there is a caveat. The resulting polygon might become concave.
To ensure that the resulting polygon remains convex, it suffices to take an area-preserving map close to the identity. Use $a=f(n),b=c=0,d=1/f(n)$ where $f(n)$ is a number dependent on $n$. The larger the value of $n$, the smaller $|1-f(n)|$ should be.

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