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.

• Question

  Is it possible to arrange all the numbers from `1` to `100` in a row such that the difference between any two adjacent numbers is at least `50`? If so, provide an example; if not, prove why.

  Topics:

  Number Theory Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample
  Sources:
  • problems.ru web site

• Question

  Is it possible to tile a `5xx5` board with dominoes?

  Note: The size of a board square matches the size of a domino square.

  Topics:

  Combinatorics -> Combinatorial Geometry Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Colorings -> Chessboard Coloring

• Question

  64 friends were told 64 news items at the same time (one news item per friend). They start calling each other and exchanging news. Each call lasts one hour. How long will it take for all the friends to know all the news? 
  א. אנג'אנס 

  Topics:

  Logic -> Reasoning / Logic
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 9-10 Question 4 Points 8

• Continue the Sequence

  `2, 6, 12, 20, 30…`

  a. What is the next number (the sixth) in the sequence?

  b. What is the `2016`th number in the sequence?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Algebra -> Sequences -> Complete/Continue the Sequence
  Sources:
  • problems.ru web site

• Question

  Can you fill a `5xx5` table with real numbers such that the sum of each row is positive, and the sum of each column is negative?

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  Topics:

  Combinatorics -> Double Counting Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction
  Sources:
  • problems.ru web site

• Question

  Can you fill a table of size `5xx5` with

  a. Integers,

  b. Real numbers,

   such that the sum of each row is even, and the sum of each column is odd?

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  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Double Counting Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction

• Question

  Given a three-digit prime number with all its digits distinct. It is known that its last digit is equal to the sum of the other two digits. Find all the possibilities for the last digit of this number.

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  Topics:

  Number Theory -> Prime Numbers Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 2, 4, and 8 Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  In two classes with an equal number of students, a quiz was administered. After grading the quiz, the teacher claimed that the number of grades of `0 ` was `13` greater than the number of all other grades combined. Is it possible that he was mistaken?

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  Topics:

  Combinatorics -> Double Counting Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction
  Sources:
  • problems.ru web site

• The Knight and the Dragon

  A knight encountered a dragon with three heads on his way and they began to fight. Every time the knight chops off one of the dragon's heads, three new heads appear in its place. Is it possible that at the end of the battle, the dragon will have a thousand heads?

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  Topics:

  Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd)

• Question

  Is it possible to make change for a `25` lira note using `10` coins worth `1`, `3`, and `5` lira?

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  Topics:

  Arithmetic Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd)
  Sources:
  • problems.ru web site