Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 4

• Question

  The plane is divided by n lines and circles.

  Prove that the resulting map can be colored with two colors such that any two adjacent regions (separated by a segment or an arc) are colored with different colors.

  Show hint

  Log in here to receive the hint
  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Induction (Mathematical Induction) Geometry -> Plane Geometry Proof and Example Combinatorics -> Colorings

• Question

  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`

  Show hint

  Log in here to receive the hint
  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Combinatorial Geometry Geometry -> Plane Geometry Geometry -> Area Calculation

• Question 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.

  Show hint

  Log in here to receive the hint
  Topics:

  Combinatorics Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Triangles -> Triangle Congruence