Problems

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

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

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 the root a of the polynomial $P(x)$ has multiplicity greater than 1 if and only if $P(a)=0$ and $P^{\prime }(a)=0$.

Prove that the following polynomial does not have any identical roots: $P(x)=1+x+x^2/2!+\cdots +x^n/n!$

For which $A$ and $B$ does the polynomial $A{x}^{n+1}+B{x}^n+1$ have the number $x=1$ at least two times as its root?

Prove that the polynomial $x^{2n}-n{x}^{n+1}+n{x}^{n-1}-1$ for $n>1$ has a triple root of $x=1$.

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 $M$.