# 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.

# Watch video

# Connected Program at Cheenta

#### Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.