Logic

Logic is the study of reasoning and valid inference. It involves analyzing statements, arguments, and deductive processes. Questions may include solving logic puzzles, evaluating the truth of compound statements, using truth tables, and identifying logical fallacies.

• How Many Liars?

  A tourist is traveling in a land of liars and truth-tellers. All truth-tellers always tell the truth, and all liars always lie.
  The tourist meets four friends: Alice, Betty, John, and Donald, and asks them: "How many of the four of you are liars?"
  Alice answers: 0
  Betty answers: 1
  John answers: 2
  Donald answers: 3
  Can we know for certain how many of them are liars?

   

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Logic -> Truth-tellers and Liars Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 1

• Donkey in the Middle

  Avi, Beni, and Gadi played "Donkey in the Middle" - at any moment someone stands in the middle and tries to catch a ball that the other two are passing. If he succeeds, one of the other two replaces him.
  After the game, it turned out that Avi stood in the middle 8 times, Beni 4 times, and Gadi 13 times.
  Who was the first and who was the last to stand in the middle?

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 3

• Doughnuts

  Ayala, Benny, Gili, Danny, and Hadas received a package of doughnuts containing

  • 10 doughnuts with dulce de leche
  • 8 doughnuts with peanut butter
  • 9 doughnuts with chocolate
  • 11 with strawberry jam

  Each of them has their favorite type of doughnut.

  • Ayala ate 5 doughnuts of her favorite type
  • Benny ate 6 doughnuts of his favorite type
  • Gili ate 7 doughnuts of her favorite type
  • Danny ate 8 doughnuts of his favorite type
  • Hadas ate 9 doughnuts of her favorite type

  After that, they were left with 3 doughnuts of different types. What is each of their favorite type of doughnut?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 5

• 6 on the Board

  The number 6 is written on the board. At each step, you can add the digit 6 to the end of the number (so that it is the units digit,) or replace the number with the sum of its digits.
  Which numbers can be obtained in this way? Describe the entire set of numbers and explain why there are no more

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Combinatorics -> Induction (Mathematical Induction) Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 6
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 5

• Another Donkey in the Middle

  Avi, Beni, and Gadi played "Donkey in the Middle" - at any given moment, someone is in the middle, trying to catch a ball that the other two are passing to each other.
  If he succeeds, one of the other two replaces him. After the game, Avi said that he was in the middle 8 times,
  Beni said he was in the middle 4 times, and Gadi forgot how many times he was in the middle.
  They also remember that Beni was the last one in the middle. Describe all the possibilities for the number of times Gadi was in the middle,

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 3

• 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

• 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

• 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

• Cards with Digits

  Rachel has three cards with different digits, all of which are greater than 0. Rachel formed all possible three-digit numbers from these cards and calculated their sum.
  Prove that the sum is divisible by 3

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Logic -> Reasoning / Logic Proof and Example
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 3-4 Question 6
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 5-6 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