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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.

#### Identical transformations ( trigonomery) , Rational and irrational numbers

The number x is such that both the sums S = sin 64x + sin 65x and C = cos 64x + cos 65x are rational numbers.
Prove that in both of these sums, both terms are rational.

#### Algebraic identities for polynomials (other) , Rational and irrational numbers

Is there a positive integer n such that

#### Identical transformations , Rational and irrational numbers

The numbers x, y and z are such that all three numbers x + yz, y + zx and z + xy are rational, and $x^2$ + $y^2$ = 1. Prove that the number $xyz^2$ is also rational.

#### Polynomials(other) , Rational and irrational numbers

Derive from the theorem in question 61013 that

is an irrational number.

#### Polynomials with integer coefficients and integer values , Rational and irrational numbers , The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers

Prove that if $(p, q) = 1$ and p/q is a rational root of the polynomial $P (x) = a_nx^n + … + a_1x + a_0$ with integer coefficients, then

a) $a_0$ is divisible by p;

b) $a_n$ is divisible by q.

#### Integer lattices (other) , Rational and irrational numbers

Prove that there is at most one point of an integer lattice on a circle with centre at $(\sqrt 2, \sqrt 3)$.

#### Polygons and polyhedra with vertices in lattice points , Rational and irrational numbers , Trigonometry (other)

A square grid on the plane and a triangle with vertices at the nodes of the grid are given. Prove that the tangent of any angle in the triangle is a rational number.

#### Identical transformations ( trigonomery) , Rational and irrational numbers

Prove that for x ≠ πn $($n is an integer$)$ sin x and cos x are rational if and only if the number tgx/2 is rational.

#### Exponential functions and logarithms (other) , Rational and irrational numbers , The fundamental theorm of arithmetic. Prime factorisation.

For what natural numbers a and b is the number $log_{a}$b rational?

#### Rational and irrational numbers , Square roots (other)

Prove that the number $\sqrt {2} + \sqrt {3} + \sqrt {5} + \sqrt {7} + \sqrt {11} + \sqrt {13} + \sqrt {17}$ is irrational.

#### Quadratic equations. Formula for the roots. , Quadratic equations. Vieta's rule. , Rational and irrational numbers

One of the roots of the equation $x^2 + ax + b = 0$ is $1 + \sqrt 3$. Find a and b if you know that they are rational.

#### Rational and irrational numbers , Surds. Rational powers (other) , Trigonometry (other)

Prove the irrationality of the following numbers:

a$)$

b$)$

+

c$)$

+

+

d$)$

e$)$ cos 10$^{\circ}$

f$)$ tg 10$^{\circ}$

g$)$ sin 1$^{\circ}$

h$)$ $log_{2}3$

#### Proof by contradiction , Rational and irrational numbers

Let the number α be given by the decimal:

a$)$ 0.101001000100001000001 …;

b$)$ 0.123456789101112131415 ….

Will this number be rational?

#### Pigeonhole principle (other) , Rational and irrational numbers

Prove that a number is rational if, and only if, it can be written as a finite or periodic decimal fraction.

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