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• Question - Apples and Pears

  There is a basket containing `30` fruits. It is known that among any `12` fruits we take from the basket, there is necessarily at least one apple, and among any `20` fruits there is necessarily one pear. How many apples and how many pears are there in the basket?

  Topics:

  Combinatorics -> Pigeonhole Principle Algebra -> Inequalities Logic -> Reasoning / Logic

• Question

  In a magical country, there are coins worth `1`, `2`, `3` and `5` liras. Yossi has `25` coins from the magical country.

  Must there necessarily be `7` coins of the same value among these coins?

  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  Find all pairs of prime numbers whose difference is `17`.

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  Topics:

  Number Theory -> Prime Numbers Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  All the even numbers from `12` to `34` are written on the board without spaces. As a result, the following number was obtained:

  `121416182022242628303234`

  Is this number divisible by `24`?

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  Topics:

  Arithmetic 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

• Question

  Is it possible for the sum of three natural numbers to be divisible by each of them?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample

• Question - Ali Baba and the Forty Thieves

  Ali Baba wrote the number `17` on a piece of paper. The forty thieves pass the paper to each other, and each one either adds `1` to the existing number, or subtracts `1`, until each of them has done so once, and then they return the paper to Ali Baba.

  Is it possible that the number now written on the paper is `40`?

  Topics:

  Arithmetic Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  Does there exist a perfect square whose digits sum to `2001`?

  Justify or provide an example!

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Number Theory -> Division -> Parity (Even/Odd) Number Theory -> Prime Numbers -> Prime Factorization Proof and Example -> Proof by Contradiction

• Question

  How many two-digit numbers are there such that the tens digit is greater than the units digit?

  Topics:

  Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  Represent the number `203` as the product of several natural numbers different from `203`, such that the sum of these numbers is also equal to `203`.

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  Topics:

  Number Theory -> Prime Numbers -> Prime Factorization

• Question

  Prove that the difference of squares of two consecutive odd numbers is divisible by `8`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities