Gillis Mathematical Olympiad, 2018-2019

• Question 1 - Children's Clubs

  In a kindergarten, there are three clubs: Judo, Agriculture, and Mathematics. Each child participates in exactly one club, and each club has at least one participant. The total number of children in the kindergarten is 32. On Friday, the kindergarten teacher gathered 6 children to tidy up the classroom. The teacher counted and found that exactly half of the Judo club members, a quarter of the Agriculture club members, and an eighth of the Mathematics club members volunteered for the task. How many students are in each club?

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  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Arithmetic -> Fractions Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division

• Question 2 - Drawing of a Relation

  Given a 5x5 grid divided into 1x1 squares. Two squares are considered related if they are in the same row or column, and the distance between their centers is 2 or 3.
  For example, in the drawing, all the squares related to the red square are marked in gray. Sammy receives a blank grid and wants to mark as many squares as possible such that no two of them are related. What is the maximum number of squares he can mark?
  

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  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry

• Question 3 - Triangles on Lines

  Six congruent isosceles triangles are arranged as shown in the figure.
  Show that points C, F, and M are collinear.

  

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  Topics:

  Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Parallel Translation / Translation