Young Mathematician Olympiad, 2019-2020, Stage B

• Question from sources: Grades 3-4(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
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 1

• Question from sources: Grades 3-4(2), Grades 5-6(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
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 2
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 1

• Question from sources: Grades 3-4(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
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 3

• Question from sources: Grades 3-4(5), Grades 5-6(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
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 5
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 2

• Question from sources: Grades 3-4(6), Grades 5-6(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
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 6
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 4

• Question from sources: Grades 3-4(1), Grades 5-6(1) - 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
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 2
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 1

• Question from sources: Grades 3-4(2), Grades 5-6(2) - 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
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 5
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 2

• Question from sources: Grades 5-6(3) - SLV LVS BLS

  In the following expression, different letters represent different digits, and identical letters represent identical digits:

  SLV = LVS + BLS

  Find the number SLV.

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rule by 11 Algebra -> Equations Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 3

• Question from sources: Grades 3-4(4), Grades 5-6(4) - 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
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 6
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 4

• Question from sources: Grades 5-6(5) - Drawing Board

  A painter has a `10 times 10` grid. Each time, the painter chooses a row or column and paints it entirely with a color of their choice.
  If they pass over a square that has already been painted with a new color, the new color completely covers the old color,
  that is, the color of the square changes.
  What is the largest number of colors we can see on this board?

  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Colorings
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 5