Problems

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 $a,b$ be positive integers and $(a,b)=1$. Prove that the quantity cannot be a real number except in the following cases $(a,b)=(1,1)$, $(1,3)$, $(3,1)$.

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 $\sqrt{2}$.

Find the largest and smallest values of the functions

a) $f_1(x)=a\cos x+b\sin x$; b) $f_2(x)=a{\cos }^{2}x+b\cos x\sin x+c{\sin }^{2}x$.

Prove that the function $\cos \sqrt{x}$ is not periodic.

The Babylonian algorithm for deducing $\sqrt{2}$. The sequence of numbers $\{x_n\}$ is given by the following conditions: $x_1=1$, $x_{n+1}=\frac{1}{2}(x_n+2/x_n)$ ($n\ge 1$).

Prove that $\underset{n\to \mathrm{\infty }}{lim}{x}_n=\sqrt{2}$.

What will the sequence from the previous problem 61297 be converging towards if we choose $x_1$ as equal to $-1$ as the initial condition?