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.

• Toys

  Jonathan has a collection of wooden toys. Some are cubes and some are spheres, some are red and some are blue.

  It is known that there are more spheres than cubes, and it is known that there are more blue toys than red toys.

  Prove that Jonathan has a blue sphere.

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 1

• 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

• Log of Wood

  You have a very long log of wood. Can you measure exactly one meter from it, if you have for this purpose:
  а. A stick with a length of one and a half meters and another stick with a length of 40 centimeters,
  б. A stick with a length of one and a half meters and another stick with a length of 30 centimeters,

  Assuming you have no other measuring tools? Explain!

  Topics:

  Combinatorics -> Invariants Algebra -> Word Problems Logic -> Reasoning / Logic Arithmetic -> Division with Remainder
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 7
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 2

• Quiz

  In a class of 25 students, a quiz was given consisting of 7 questions. Prove that at least one of the following two statements is true:

  1. There is a student who solved an odd number of questions.
  2. There is a question that was solved by an even number of students.
  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 5

• Circle of Liars - The Truth Claim

  In a circle, `n` people are seated, each of whom is either a liar or a truth-teller.

  The people are looking towards the center of the circle. A liar always lies, and a truth-teller always tells the truth.

  Each of the people knows exactly who is a liar and who is a truth-teller.

  Each of the people says that the person sitting two places to their left (that is, next to the person sitting next to them), is a truth-teller.

  It is known that in the circle there is at least one liar, and at least one truth-teller.

  a. Is it possible that `n = 2017`?

  b. Is it possible that `n = 5778`?

  (Solution format: "word, word" for example "cat, puppy")

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  Topics:

  Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction Logic -> Truth-tellers and Liars Problems
  Sources:
  • Gillis Mathematical Olympiad, 2017-2018 Question 1

• Number Network

  In the diagram, the numbers on the edges indicate the differences between the numbers inside the circles. Place positive numbers inside the circles and discover what the number in the bottom circle is.

  

  Topics:

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

• Question

  A two-digit number is written on the board.
  Avi said: "The units digit of the number is 3"
  Beni said: "It's a square number"
  Gili said: "This number is a multiple of 12"
  Then the teacher said: "There are two correct statements and one wrong one here."
  What number was written on the board?

  Show hint

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Logic -> Truth-tellers and Liars Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 1

• Question

  Vered subtracted a number composed of the same digits written in reverse order from a three-digit number.
  As a result, she obtained a two-digit number. Find the two-digit number she obtained.

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Algebra Logic -> Reasoning / Logic
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 2

• Question

  A two-digit number is written on the board.
  Avi said: "The digit 5 appears in this number."
  Beni said: "This is a square number."
  Gili said: "This number is greater than 50."
  Dani said: "The number is divisible by 7."
  Then the teacher said: "There are three correct statements here and one incorrect statement.".
  What number was written on the board?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 1

• Minimal Table

  Given a table of size `3 times 3`. Hilla wants to write digits from 1 to 9 in the table's cells, such that all the sums in the rows and columns of the table are different, and the total sum of the table is as small as possible.
  It is allowed to repeat the same digit multiple times. What is the smallest sum that Hilla can obtain?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Number Tables
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 7