Select Page

# 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
##### Suggested Book
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.

# Connected Program at Cheenta

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.

# Similar Problems

## Lines and Angles | PRMO 2019 | Question 7

Try this beautiful problem from the Pre-RMO, 2019 based on Lines and Angles. You may use sequential hints to solve the problem.

## Logarithm and Equations | AIME I, 2012 | Question 9

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2015 based on Logarithm and Equations.

## Cross section of solids and volumes | AIME I 2012 | Question 8

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Cross section of solids and volumes.

## Angles of Star | AMC 8, 2000 | Problem 24

Try this beautiful problem from GeometryAMC-8, 2000 ,Problem-24, based triangle. You may use sequential hints to solve the problem.

## Unit digit | Algebra | AMC 8, 2014 | Problem 22

Try this beautiful problem from Algebra about unit digit from AMC-8, 2014. You may use sequential hints to solve the problem.

## Problem based on Integer | PRMO-2018 | Problem 6

Try this beautiful problem from Algebra based on Quadratic equation from PRMO 8, 2018. You may use sequential hints to solve the problem.

## Number counting | ISI-B.stat Entrance | Objective from TOMATO

Try this beautiful problem Based on Number counting .You may use sequential hints to solve the problem.

## Area of a Triangle | AMC-8, 2000 | Problem 25

Try this beautiful problem from Geometry: Area of the triangle from AMC-8, 2000, Problem-25. You may use sequential hints to solve the problem.

## Mixture | Algebra | AMC 8, 2002 | Problem 24

Try this beautiful problem from Algebra based on mixture from AMC-8, 2002.. You may use sequential hints to solve the problem.

## Trapezium | Geometry | PRMO-2018 | Problem 5

Try this beautiful problem from Geometry based on Trapezium from PRMO , 2018. You may use sequential hints to solve the problem.