Combinatorics, Case Analysis / Checking Cases, Processes / Procedures

This category covers problems involving sequences of operations or steps that evolve over time or iterations. Questions might ask about the outcome of a process, whether it terminates, or properties of its state after a certain number of steps. Often related to algorithms or invariants.

• 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

• How Many Triangles - 2?

  How many triangles are in the picture?

  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 6

• 5 Lines, 8 Intersections

  Draw 5 lines such that there are exactly 8 points of intersection between them.

  Topics:

  Combinatorics -> Combinatorial Geometry Geometry -> Plane Geometry Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 3-4 Question 1

• Who Likes Cola?

  Shmuel has 4 children: Ohad, Benny, Guy, and Dor. Each of them has one drink that he likes (no two people share a favorite drink): water, cola, grape juice, and orange juice.
  In addition, each of them also has a favorite subject (and no two have the same one): mathematics, physics, computer science, and chemistry.

  Hints:

  1. 1. Ohad likes computer science
  2. 2. The one who likes mathematics does not drink orange juice
  3. 3. Dor does not like physics
  4. 4. Benny drinks water
  5. 5. Guy does not drink grape juice
  6. 6. The child who likes chemistry drinks grape juice
  7. 7. Benny does not like mathematics

  Question: Who likes cola?

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 3-4 Question 2

• Compose a Sum

  Compose three numbers from the digits `1,2,3,4,5,6,7,8,9` such that one of them is the sum of the other two.
  Each digit must be used exactly once.

  Topics:

  Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 3-4 Question 4
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 5-6 Question 1

• How Many Sums Are Possible?

  What is the number of possible results of addition exercises of the form `A + B`
  where A and B are two distinct positive single-digit numbers?

  Topics:

  Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 3-4 Question 5

• We went on a trip

  Grade 4, consisting of 32 students, went on a trip. The students had to bring hats, sunglasses, and water bottles.
  No child forgot all of these things, but:

  • Among the students who brought hats, 9 forgot sunglasses,
  • Among the students who brought sunglasses, 7 forgot water bottles,
  • And among the students who brought water bottles, 10 forgot sunglasses.

  How many students in the class brought everything needed for the trip?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Set Theory Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 5-6 Question 2

• Colorful Street

  Along the street are 16 houses, in red, blue, and green. There is at least one house of each color. No two adjacent houses are of the same color.
  Between any two blue houses there is a red house. Between any two green houses there is a blue house and a red house.
  What is the largest possible number of green houses?
  Note: The street is straight, all houses are located on one side of the street.

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 3-4 Question 6

• Colorful Street 2

  There are 15 houses along the street, colored red, blue, and green. There is at least one house of each color.
  Between any two blue houses there is a red house. Between any two green houses there is a blue house.
  What is the largest possible number of green houses?
  Note: The street is straight, and all houses are located on one side of the street.

  Topics:

  Combinatorics -> Pigeonhole Principle Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 5-6 Question 5
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 6

• Magic Fractions

  Let's call a fraction magic if both its numerator and denominator are less than 10. For example, the fraction `1/9` is considered magic, the fraction `6/8` is also magic, but the fraction `3/14` is not magic.
  How many magic fractions are there that are greater than one-half and less than 1?

  Note: For the purpose of this question, `2/3` and `4/6` are considered different fractions.

  Topics:

  Arithmetic -> Fractions Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 5-6 Question 6