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• Question

  We call a number nice if it consists only of odd digits. How many four-digit nice numbers exist?

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  Topics:

  Arithmetic Combinatorics -> Product Rule / Rule of Product

• Question

  In a foreign country, every phone number has `7` digits, and the first digit is always different from `0`. What is the maximum number of phone numbers that can exist in that country?

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  Topics:

  Arithmetic Combinatorics -> Product Rule / Rule of Product

• Question

  In a country there are `100` cities, and from each city `4` roads lead out (a road can only start or end in a city). How many roads are there in this country?

  Topics:

  Combinatorics -> Double Counting Combinatorics -> Graph Theory

• Question

  The sum of six distinct natural numbers is equal to `22`. Find these numbers.

  Topics:

  Number Theory Arithmetic

• 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

• Question

  Someone made `15` point-like holes in a carpet that is `4xx4` meters in size. Is it always possible to cut out a rug of size `1xx1` meter from the original carpet such that it has no holes?

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  Topics:

  Geometry Combinatorics -> Pigeonhole Principle Combinatorics -> Combinatorial Geometry

• Question

  There are `85` balloons in a room, red and blue. It is known that:

  - At least one of the balloons is red,

  - In any pair of balloons we take, at least one of the balloons must be blue.

  How many red balloons are in the room?

  Topics:

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

• Question

  a. The apple is heavier than the peach, and the peach is heavier than the pear. Which is heavier – the apple or the pear?

  b. The orange is heavier than the pear, and the plum is lighter than the orange. Which is heavier – the pear or the plum?

  Topics:

  Arithmetic Algebra -> Inequalities Logic -> Reasoning / Logic

• Question

  Three hedgehogs have three pieces of cheese weighing `5`, `8`, and `11` grams. A fox offers to help the hedgehogs divide the cheese equally. The fox can bite off one gram from each of two cheese pieces of its choice. Can the fox, using these actions, reach a state where it leaves the three hedgehogs with equal pieces of cheese?

  Topics:

  Arithmetic Algebra -> Word Problems Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division

• Question

  A tourist arrived at an island inhabited by truth-tellers and liars. Truth-tellers always tell the truth, and liars always lie. The tourist is traveling with a local guide, and they see a farmer working in the field. The tourist asked the guide to find out if this man is a truth-teller or a liar. The guide spoke with the farmer and then said to the tourist: "He said that he is a truth-teller."

  A. What can be deduced about the farmer?

  B. What can be deduced about the guide?

  Topics:

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