Young Mathematician Olympiad, 2019-2020

• Question from sources: Stage A, Grades 5-6(4) - Five Numbers

  Given five distinct positive integers. The sum of these numbers is 27. Additionally, it is known that the product of these five numbers is odd. Calculate this product.

  Topics:

  Arithmetic Number Theory -> Division -> Parity (Even/Odd)
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6 Question 4

• Question from sources: Stage A, Grades 5-6(5) - Angles

  Calculate the sum of the marked angles:

  

  Topics:

  Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Angle Calculation
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6 Question 5

• Question from sources: Stage A, Grades 5-6(6) - More Game Cubes

  Aviv has game cubes, where two opposite faces of each are painted red and the rest are blue.
  Aviv glued together a `3 xx 3 xx 3` cube from the game cubes. Then his friend Kfir came and calculated the total red area on the surface of the large cube.
  What is the largest result Kfir can get?

  Topics:

  Geometry -> Solid Geometry / Geometry in Space Combinatorics -> Combinatorial Geometry Minimum and Maximum Problems / Optimization Problems Combinatorics -> Colorings
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6 Question 6

• Question from sources: Stage A, Grades 5-6(7) - More Numbers with Ascending Digits

  Miri writes down all the numbers whose decimal representation contains only the digits `1, 2, 3, 4, 5, 6`.
  (Not all digits must appear) and all the digits that appear are written in ascending order (e.g., 1356 or 124 or 5 but not 162 and not 1223).
  How many numbers will Miri write down?

  Topics:

  Combinatorics -> Product Rule / Rule of Product Set Theory
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6 Question 7

• Question from sources: Stage B, 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: Stage B, Grades 3-4(2), Stage B, 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: Stage B, 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: Stage B, Grades 3-4(5), Stage B, 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: Stage B, Grades 3-4(6), Stage B, 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: Stage B, Grades 3-4(1), Stage B, 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