Mink Exercises and Additional Competition Materials, 2018-2019

Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise 5 Exercise 6 Exercise 7 Exercise 8

• Question from sources: Exercise 4

  Inside a square with side length 1, `n>=101` points are marked, such that no three are collinear. A triangle is called marked if its vertices are marked points. Prove that the area of one of the marked triangles is less than `1/100`

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  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Combinatorial Geometry Geometry -> Plane Geometry Geometry -> Area Calculation
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 4

• Question from sources: Exercise 4(3)

  Given a regular polygon with n vertices. Calculate the number of distinct (non-congruent) triangles whose vertices coincide with the vertices of the polygon.

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  Topics:

  Combinatorics Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Triangles -> Triangle Congruence
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 4 Question 3