Prove Existence of 5 & 64 Points in Plane with 8 & 2005 Right-Angled Triangles

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SUMMARY

This discussion addresses the geometric problem of proving the existence of specific configurations of points in the plane that yield a designated number of right-angled triangles. Specifically, it establishes that there are 5 points that can form 8 right-angled triangles and 64 points that can form at least 2005 right-angled triangles. The problem is sourced from the Junior Balkan Mathematical Olympiad 2005, indicating its relevance in competitive mathematics.

PREREQUISITES
  • Understanding of combinatorial geometry
  • Familiarity with properties of right-angled triangles
  • Knowledge of point configuration in Euclidean space
  • Basic principles of mathematical proofs
NEXT STEPS
  • Research combinatorial geometry techniques for point configurations
  • Study the properties and theorems related to right-angled triangles
  • Explore mathematical proof strategies in geometry
  • Investigate previous problems from the Junior Balkan Mathematical Olympiad
USEFUL FOR

Mathematicians, geometry enthusiasts, students preparing for mathematical competitions, and educators looking for challenging problems in combinatorial geometry.

sachinism
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Prove that there exist

(a) 5 points in the plane so that among all the triangles with vertices among these points there are 8 right-angled ones;

(b) 64 points in the plane so that among all the triangles with vertices among these points there are at least 2005 right-angled ones.
 
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CRGreathouse said:
Source: Junior Balkan MO 2005

hey thanks for the website... looks good

thanks again
 

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