# Understand the problem

Let be the area of the triangle . A non-regular convex polygon is called

*guayaco*if exists a point in its interior such thatShow that, for every integer , a guayaco polygon of sides exists.##### Source of the problem

Cono sur olympiad 2017

##### Topic

Geometry

##### Difficulty Level

Easy

##### Suggested Book

Problem Solving Strategies by Arthur Engel.

# Start with hints

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 . Show that, if 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 -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 where is a number dependent on . The larger the value of , the smaller should be.

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#### Math Olympiad Program

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