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#### Division with remainder , Periodicity and aperiodicity , Probability theory (other) , Proof by exhaustion

A numerical sequence is defined by the following conditions:

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Prove that among the terms of this sequence there are an infinite number of complete squares.

#### Division with remainder , Periodicity and aperiodicity , Probability theory (other) , Proof by exhaustion

A numerical sequence is defined by the following conditions:
$\\$

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How many complete squares are found among the first members of this sequence, not exceeding 1,000,000?

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

The numbers a and b are such that the first equation of the system

$cos x = ax + b$

$sin x + a = 0$

has exactly two solutions. Prove that the system has at least one solution.

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

The numbers a and b are such that the first equation of the system

$sin x + a = bx$

$cos x = b$

has exactly two solutions. Prove that the system has at least one solution.

#### Odd and even numbers , Polynomials with integer coefficients and integer values , Rational and irrational numbers

n numbers are given as well as their product, p. The difference between p and each of these numbers is an odd number.
Prove that all n numbers are irrational.

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