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#### Division with remainder , Pigeonhole principle (other)

Prove that amongst the numbers of the form $19991999…19990…0$ – that is 1999 a number of times, followed by a number of 0s – there will be at least one divisible by 2001.

#### Division with remainder , Pigeonhole principle (other)

Prove that in any group of 2001 whole numbers there will be two whose difference is divisible by 2000.

#### Pigeonhole principle (other) , Proof by contradiction

In a group of seven boys, everyone has at least three brothers who are in that group. Prove that all seven are brothers.

#### Decimal number system , Pigeonhole principle (other)

7 different digits are given. Prove that for any natural number n there is a pair of these digits, the sum of which ends in the same digit as the number.

#### Pigeonhole principle (other)

Prove that from any 27 different natural numbers less than 100, two numbers that are not coprime can be chosen.

#### Pigeonhole principle (other) , The fundamental theorm of arithmetic. Prime factorisation.

The product of a group of 48 natural numbers has exactly 10 prime factors. Prove that the product of some four of the numbers in the group will always give a square number.

#### Pigeonhole principle (other) , The fundamental theorm of arithmetic. Prime factorisation.

The product of 1986 natural numbers has exactly 1985 different prime factors. Prove that either one of these natural numbers, or the product of several of them, is the square of a natural number.

#### Arithmetic of remainders , Pigeonhole principle (other)

The sum of 100 natural numbers, each of which is no greater than 100, is equal to 200. Prove that it is possible to pick some of these numbers so that their sum is equal to 100.

#### Pigeonhole principle (other) , Proof by exhaustion

Is it possible to place the numbers $1, 2,…12$ around a circle so that the difference between any two adjacent numbers is 3, 4, or 5?

#### Decimal number system , Pigeonhole principle (other)

What is the maximum difference between neighbouring numbers, whose sum of digits is divisible by 7?

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