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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?

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.

Author: A.K. Tolpygo

An irrational number α, where 0 $<$α $<$½, is given. It defines a new number $α_1$ as the smaller of the two numbers 2α and 1 - 2α. For this number, $α_2$ is determined similarly, and so on.

a) Prove that for some n the inequality $α_n <3/16$ holds.

b) Can it be that $α_n> 7/40$ for all positive integers n?

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.

Author: A.V. Shapovalov

We call a triangle rational if all of its angles are measured by a rational number of degrees. We call a point inside the triangle rational if, when we join it by segments with vertices, we get three rational triangles. Prove that within any acute-angled rational triangle there are at least three distinct rational points.

In the Republic of mathematicians, the number α $>$ 2 was chosen and coins were issued with denominations of 1 pound, as well as in $α^k$ pounds for every natural k. In this case α was chosen so that the value of all the coins, except for the smallest, was irrational. Could it be that any amount of a natural number of pounds can be made with these coins, using coins of each denomination no more than 6 times?

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.

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.

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

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.

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.

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

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.

Is it possible for

a$)$ the sum of two rational numbers irrational?

b$)$ the sum of two irrational numbers rational?

c$)$ an irrational number with an irrational degree to be rational?

Prove the irrationality of the following numbers:

a$)$

b$)$

+

c$)$

+

+

d$)$

–

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

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

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

h$)$ $log_{2}3$

Let the number α be given by the decimal:

a$)$ 0.101001000100001000001 …;

b$)$ 0.123456789101112131415 ….

Will this number be rational?

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