Problems

In the number $a=0.12457\dots$ the $n$th digit after the decimal point is equal to the digit to the left of the decimal point in the number. Prove that $\alpha$ is an irrational number.

With a non-zero number, the following operations are allowed: $x\to \frac{1+x}{x}$, $x\to \frac{1-x}{x}$. Is it true that from every non-zero rational number one can obtain each rational number with the help of a finite number of such operations?

At all rational points of the real line, integers are arranged. Prove that there is a segment such that the sum of the numbers at its ends does not exceed twice the number on its middle.

A numeric set $M$ containing 2003 distinct numbers is such that for every two distinct elements $a,b$ in $M$, the number $a^2+b\sqrt{2}$ is rational. Prove that for any $a$ in $M$ the number $q\sqrt{2}$ is rational.

Ten pairwise distinct non-zero numbers are such that for each two of them either the sum of these numbers or their product is a rational number.

Prove that the squares of all numbers are rational.

Prove that the infinite decimal $0.1234567891011121314\dots$ (after the decimal point, all of the natural numbers are written out in order) is an irrational number.

Let the number $\alpha$ be given by the decimal:

a) $0.101001000100001000001\dots$;

b) $0.123456789101112131415\dots$.

Will this number be rational?

Prove the irrationality of the following numbers:

a) $\sqrt{3}17$

b) $\sqrt{2}+\sqrt{3}$

c) $\sqrt{2}+\sqrt{3}+\sqrt{5}$

d) $\sqrt{3}3-\sqrt{2}$

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

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

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

h) ${\log }_{2}3$

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?

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.