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.

• Camels and Horses

  In a herd, there are dromedary camels (one-humped), Bactrian camels (two-humped), and horses. It is known that the total number of humps in this herd is `200`. In addition, the number of horses is equal to the number of Bactrian camels. How many animals are there in the herd in total?

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic

• Milk and Coffee

  A. Hannah has two cups, one with milk and the other with coffee. The amounts of milk and coffee are the same. Hannah transfers two teaspoons of milk into the cup with coffee, mixes, and then transfers two teaspoons from the cup of coffee into the milk. Currently, she has two cups each containing a mixture of coffee and milk. What is greater – the amount of milk in the cup of coffee or the amount of coffee in the cup of milk?

  B. The same question, but this time Hannah adds milk to the coffee without mixing.

  Topics:

  Combinatorics -> Invariants Logic -> Reasoning / Logic

• Question

  Hannah is waiting for a bus. Which of the following three events is most likely to occur:

  1. Hannah waits for the bus for at least one minute,
  2. Hannah waits for the bus for at least two minutes,
  3. Hannah waits for the bus for at least five minutes.
  Topics:

  Logic -> Reasoning / Logic Set Theory Probability Theory

• Watering Potted Plants

  Hannah and Rachel watered potted plants at home. It is known that each of them watered exactly half the number of potted plants in the house, and it is also known that three potted plants were accidentally watered twice. How many potted plants remain to be watered?

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Set Theory

• Knights of the Round Table

  Around a round table sit 12 knights, each of whom is either an elf or a dwarf. It is known that the number of elves is greater than the number of dwarves. Prove that there are two elves who sit opposite each other.

  Will this continue to be true if the total number of knights is 120?

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction

• Question

  There are `30` students in a class. During a test, Pinchas made `13` mistakes, and the rest made fewer mistakes. Prove that there are three students who made the same number of mistakes.

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction Minimum and Maximum Problems / Optimization Problems

• Who Broke the Glass?

  One of the following four students: Avi, Benny, Gili, and Danny – broke a glass. The principal asks them who did it. Here are the answers she received:

  Avi: I know for certain that whoever broke the glass, it was not me or Benny.

  Benny: I know for certain that whoever broke the glass, it was not me or Danny.

  Gili: I know for certain that whoever broke the glass, it was not me or Benny.

  Danny: I know for certain that whoever broke the glass, it was not me or Avi.

  It is known that only one of them broke the glass, and it is also known that three of the students told the truth, and one lied. So who broke the glass?

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Logic -> Truth-tellers and Liars Problems

• Question

  This year, Yossi participated in the Mathematics Olympiad for the first time. Before the Olympiad, a survey was conducted in which each participant was asked what place they thought they would achieve. Yossi answered that he would probably be in last place. After the competition itself, it turned out that every participant, except for Yossi, achieved a worse place than they answered in the survey. What place did Yossi take in the Olympiad?

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic

• Question

  A two-digit number is written on the board. Avi claims that the units digit of the number is twice the tens digit. Beni claims that the number is divisible by `9`. Gal claims that the number is divisible by `4`. Dani claims that the number is divisible by `27`. It is known that one of them is wrong, and the rest are correct. What number is written on the board?

  Topics:

  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 Algebra -> Word Problems Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  How many triangles are there in the image?

  

  Topics:

  Geometry -> Plane Geometry -> Triangles Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures