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.

• Question

  Every evening, Yuval finishes work at a random time and arrives at a bus stop. At this station, two buses stop: number `7`, which goes to Yuval's house, and number `13`, which goes to the house of his friend Shlomi. Yuval gets on the first bus that arrives and, depending on that, goes to Shlomi's or home.

  After a while, Yuval notices that after work, he goes to Shlomi's about twice as often as he goes home. He deduced from this that bus number `13` arrives twice as frequently as bus number `7`.

  Is Yuval necessarily correct?

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  Topics:

  Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Probability Theory Logic -> Reasoning / Logic -> Paradoxes

• Baobab

  In the following exercise, identical digits have been replaced with identical letters, and different digits have been replaced with different letters. Reconstruct the exercise.

  `BAOxxBAxxB = 2002`

   

   

   

   

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Number Theory -> Prime Numbers -> Prime Factorization Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic

• Cats for Grandma

  Grandma Hannah has a number of cats. One day, her three grandchildren, Avi, Benny, and Gili, came to visit and tried to guess how many cats she has.

  Avi said: "Grandma has at least 7 cats",

  Benny said: "But less than 10, I think",

  Then Gili said: "Grandma Hannah has either 9, or 10, or 11 cats".

  "You are all correct!" replied the grandma

   So how many cats does she have?
   

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  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 4

• Palindromic Number

  Find a four-digit palindromic number that is divisible by 25 and not divisible by 3.

  Note: A palindromic number is a number that does not change if its digits are read in reverse order. For example, the number 5775 is a palindromic number, and the number 5778 is not a palindromic number.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Number Theory -> Division -> Parity (Even/Odd) Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 5 and 25 Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 5

• What is in each container?

  On the table, a cup, glass, pitcher, and jar are arranged in a row in an unknown order. The containers hold milk, orange juice, cola, and water, but it is unknown which liquid is in each container. Given that:

  • The milk and water are not in the cup.
  • The container holding orange juice is between the pitcher and the container holding cola.
  • The jar does not contain water or orange juice.
  • The glass is between the jar and the container with the milk.

  Which liquid is in which container?

   

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  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Number Tables

• Question

  Grandma Hannah has many flowerpots with flowers in her house. One day, her three grandchildren, Avi, Beni, and Gili, came to visit and tried to guess how many flowerpots she has.
  Avi said: "Grandma has more than 8 flowerpots,"
  Beni said: "More than 10, I think,"
  And then Gili said: "Grandma Hannah has at least 12 flowerpots."
  "Two of you are right, and one of you is wrong," answered the grandmother. So how many flowerpots does she have in the house?

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 3

• The Magic Octopuses

  In the magic sea live octopuses who can talk. Each octopus either always tells the truth or always lies. One day
  the following conversation took place between four octopuses, Avi, Benny, Gidi, and Danny:

  
  Avi: I am a green octopus
  Benny: I am not green
  Gidi: All green octopuses are liars
  Danny: Only a green octopus can be a liar

  
  It is known that only one of these four is a liar, and the rest are truthful.
  a. Who is the liar among the four friends? Explain!
  b. Is it possible to know what his color is?

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Logic -> Truth-tellers and Liars Problems
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 6
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 4

• Pumbaa and the Candies

  Pumbaa has 11 chocolate candies and 13 toffee candies. Each time he can eat either two candies of different types,
  or three candies of the same type. What is the largest number of candies that Pumbaa can eat according to these rules?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Combinatorics -> Invariants Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 6

• Multiplying Rooks

  Given an `n times n` board where each square contains either 1 or -1.

  Define the value of an arrangement of rooks on the board as the product of the numbers on their squares.

  Is the sum of the values of all arrangements necessarily divisible by 4?

  Also given `n >= 4`

  (Solution for verification: yes or no)

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  Topics:

  Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Digit Sum - 9

  How many numbers in the range from 1 to 500 have a digit sum of 9?

  Topics:

  Number Theory Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage A, Grades 3-4 Question 2