Problems

Prove that for any natural number there is a multiple of it, the decimal notation of which consists of only 0 and 1.

Without calculating the answer to $2^{30}$, prove that it contains at least two identical digits.

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

a) $0.101001000100001000001\dots$;

b) $0.123456789101112131415\dots$.

Will this number be rational?

Prove that there are infinitely many composite numbers among the numbers $\lfloor 2^k\sqrt{2}\rfloor$ ($k=0,1,\dots$).

Prove that if $(m,10)=1$, then there is a repeated unit $E_n$ that is divisible by $m$. Will there be infinitely many repeated units?

Is it possible to draw from some point on a plane $n$ tangents to a polynomial of $n$-th power?

It is known that $\cos {\alpha }^{\circ }=1/3$. Is $\alpha$ a rational number?

Let $f(x)$ be a polynomial of degree $n$ with roots ${\alpha }_1,\dots ,{\alpha }_n$. We define the polygon $M$ as the convex hull of the points ${\alpha }_1,\dots ,{\alpha }_n$ on the complex plane. Prove that the roots of the derivative of this polynomial lie inside the polygon $M$.

For what values of $n$ does the polynomial $(x+1{)}^n-x^n-1$ divide by:

a) $x^2+x+1$; b) $(x^2+x+1{)}^2$; c) $(x^2+x+1{)}^3$?

a) Using geometric considerations, prove that the base and the side of an isosceles triangle with an angle of ${36}^{\circ }$ at the vertex are incommensurable.

b) Invent a geometric proof of the irrationality of $\sqrt{2}$.