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

Are there such irrational numbers a and b so that a $>$ 1, b $>$ 1, and [$a^m$] is different from [$b^n$] for any natural numbers m and n?

#### 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.

#### Odd and even numbers

Are the sum and product odd or even for:

a$)$ two even numbers?

b$)$ two odd numbers?

c$)$ an odd and an even number?

#### Examples and counterexamples. Constructive proofs , Functions. Continuity (other) , Odd and even numbers

Are there functions p $($x$)$ and q $($x$)$ such that p $($x$)$ is an even function and p $($q $($x$)$$)$ is an odd function $($different from identically zero$)$?

#### Odd and even numbers

Let m and n be integers. Prove that mn$($m + n$)$ is an even number.

#### Odd and even numbers

Could the difference of two integers multiplied by their product be equal to the number 1999?

#### Odd and even numbers

Around a table sit boys and girls. Prove that the number of pairs of neighbours of different sexes is even.

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