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?

Find all functions $f(x)$ defined for all positive $x$, taking positive values and satisfying the equality $f(x^y)=f(x{)}^f(y)$ for any positive $x$ and $y$.

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.

Does there exist a real number $\alpha$ such that the number $\cos \alpha$ is irrational, and all the numbers $\cos 2\alpha$, $\cos 3\alpha$, $\cos 4\alpha$, $\cos 5\alpha$ are rational?

Prove that if the irreducible rational fraction $p/q$ is a root of the polynomial $P(x)$ with integer coefficients, then $P(x)=(qx-p)Q(x)$, where the polynomial $Q(x)$ also has integer coefficients.

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.

Are there any irrational numbers $a$ and $b$ such that the degree of $a^b$ is a rational number?