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Problem 1

Prove that among any 39 consecutive natural numbers

it is always possible to find one whose sum of digits

is divisible by 11.

Problem 2

Sets of 4 positive numbers are made out of each other according

to the following rule: (a, b, c, d) (ab, bc, cd, da).

Prove that in this (infinite) sequence (a, b, c, d) will

never appear again, except when a = b = c = d = 1.

Problem 3

Take a series of the numbers 1 and (-1) with a length

of 2k (k is natural). The next set is made by multiplying

each number by the next one; the last is multiplied by the

first. Prove that eventually the set will contain only ones.

Problem 4

What is the largest x for which

427 + 41000 + 4x

equals the square of a whole number?

Problem 5

Prove that for any prime number p > 2 the numerator m of the fraction

http://

is divisible by p.

Prove that among any 39 consecutive natural numbers

it is always possible to find one whose sum of digits

is divisible by 11.

Problem 2

Sets of 4 positive numbers are made out of each other according

to the following rule: (a, b, c, d) (ab, bc, cd, da).

Prove that in this (infinite) sequence (a, b, c, d) will

never appear again, except when a = b = c = d = 1.

Problem 3

Take a series of the numbers 1 and (-1) with a length

of 2k (k is natural). The next set is made by multiplying

each number by the next one; the last is multiplied by the

first. Prove that eventually the set will contain only ones.

Problem 4

What is the largest x for which

427 + 41000 + 4x

equals the square of a whole number?

Problem 5

Prove that for any prime number p > 2 the numerator m of the fraction

http://

is divisible by p.

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