Solving a 1415-gon Triangle Problem with the Pigeonhole Principle

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In summary, the conversation is about a mathematical problem involving a convex polygon of 1415 sides and its circumference of 2001 cm. The goal is to prove that there are 3 vertices that form a triangle with an area less than 1 cm2. The conversation discusses various approaches to solve the problem, including using the sum of interior and exterior angles, the average length of a side, and the Pigeonhole principle. The participants also mention the possibility of using induction and a direct proof, with a potential for a smaller bound. One participant suggests a possible approach involving calculating the area at each corner using the lengths of adjacent sides and exterior angles.
  • #1
Numeriprimi
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Hello. I have an example for you. I'm curious how. Yesterday I was on the mathematical competition. One example I can not solve. I want to know how. Can you help me, please?

Consider a convex polygon of 1415 sides, which circumference is 2001 cm. Prove that between its peaks, there are 3 such that vertices form a triangle with an area less than 1 cm2.

So... We know it is 1415-gon. The sum of its interior angles is π(n-2) rad (where n is 1415). We can calculate the average length of a side: 2001/1415 cm...
And the last what I think: there is Pigeonhole principle but i don't know how to do it.

Thanks very much for your ideas and sorry for my bad English.
 
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  • #2
This is so extremely close to sqrt(2) that I don't think it is a coincidence. As the corresponding problem for n=4 sides is exact, I would expect that induction is possible.
 
  • #3
Hmmm... Where is sqrt(2)?
 
  • #4
2001/1415 = 1.414134...
sqrt(2) = 1.414214...
and 2001/1415 < sqrt(2) < 2002/1415

Edit: Forget induction, there are polygons which would not be covered there.

I found a direct proof, and there is a good safety margin - it is possible to prove a better bound (smaller than 0.1cm^2).
 
Last edited:
  • #5
Well, still do not know how to do it ... I don't understand why.
 
  • #6
It is probably not the intended way to solve it (as it does not use this sqrt(2)-relation), but here is a possible approach:

For any corner, can you calculate the corresponding area as function of the lengths of the adjacent sides and the exterior angle at this corner? This relation is true for all corners and you know something about the sum of those exterior angles.
 

1. How does the Pigeonhole Principle apply to solving a 1415-gon triangle problem?

The Pigeonhole Principle states that if there are more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon. This principle can be applied to the 1415-gon triangle problem by considering each vertex of the polygon as a pigeonhole and each side of the polygon as a pigeon. This allows us to use the principle to find relationships between the number of sides, vertices, and angles of the polygon.

2. What is a 1415-gon triangle?

A 1415-gon triangle is a polygon with 1415 sides that can be divided into triangles. It is a unique geometric shape that poses interesting mathematical challenges and has practical applications in fields such as architecture, engineering, and computer graphics.

3. What are some real-life applications of solving a 1415-gon triangle problem?

The Pigeonhole Principle and its application to solving the 1415-gon triangle problem have various real-life applications. For example, it can be used in computer graphics to optimize the rendering of complex shapes, in cryptography to analyze the efficiency of encryption algorithms, and in scheduling problems to determine the minimum number of time slots required to schedule a set of tasks with different durations.

4. Are there any limitations to using the Pigeonhole Principle to solve the 1415-gon triangle problem?

While the Pigeonhole Principle is a useful tool in solving the 1415-gon triangle problem, it has its limitations. It only provides a theoretical solution and does not necessarily consider all possible scenarios. Also, it assumes that all the pigeons (or sides) are identical, which may not always be the case in real-life applications.

5. How can the Pigeonhole Principle be extended to solve other geometric problems?

The Pigeonhole Principle can be extended to solve a wide range of geometric problems, such as those involving polygons with different numbers of sides or shapes other than triangles. It can also be used in combination with other mathematical principles and techniques to solve more complex problems in geometry, number theory, and combinatorics.

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