Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4

• Question 1 - The Round Table

  Around a round table are 12 chairs, with knights sitting on some of them. Arthur wants to join the meeting,
  and it turns out that no matter where he sits, someone is definitely sitting next to him.
  What is the smallest number of knights that can be around the table to ensure this is true? (not including Arthur)

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question 2 - Two Hashes

  What is the maximum number of "domino" shapes (rectangles `1 times 2` or `2 times 1`) that can be placed inside the orange shape,
  such that they do not overlap and do not extend beyond the boundaries of the shape?

  

  Topics:

  Combinatorics -> Invariants Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems Combinatorics -> Colorings -> Chessboard Coloring Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Additional sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 1

• Question 3 - Domino Tiles

  A domino tile is a rectangle composed of two squares, with each square marked with dots.
  The number of dots on each square can be from 0 to 6, and each pair of numbers appears exactly once (regardless of order).
  In total, there are 28 tiles in the game. How many dots are there on these tiles in total?

  Topics:

  Arithmetic Combinatorics

• Question 5 - The Bakery

  In the morning, a bakery's storage room contained 135 kilograms of flour and 92 kilograms of sugar.
  To bake one cake, the baker uses one kilogram of flour and one kilogram of sugar.
  At the end of the workday, the amount of flour remaining for the baker was twice as large as the amount of sugar remaining.
  How many cakes did the baker make during the workday?

  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Additional sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 2

• Question 6 - The Number

  Given a positive integer less than 2000.
  If it is not divisible by 43, then it is divisible by 41,
  If it is not divisible by 53, then it is divisible by 43,
  If it is not divisible by 41, then it is divisible by 53.
  Find the number.

  Topics:

  Number Theory -> Prime Numbers Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division
  Additional sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 4