Problems

Prove that if $(p,q)=1$ and $p/q$ is a rational root of the polynomial $P(x)=a_n{x}^n+\cdots +a_{1}x+a_0$ with integer coefficients, then

a) $a_0$ is divisible by $p$;

b) $a_n$ is divisible by $q$.

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?

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.

Old calculator I.

a) Suppose that we want to find $\sqrt[3]{x}$ ($x>0$) on a calculator that can find $\sqrt{x}$ in addition to four ordinary arithmetic operations. Consider the following algorithm. A sequence of numbers $\{y_n\}$ is constructed, in which $y_0$ is an arbitrary positive number, for example, $y_0=\sqrt{\sqrt{x}}$, and the remaining elements are defined by $y_{n+1}=\sqrt{\sqrt{x{y}_n}}$ ($n\ge 0$).

Prove that $\underset{n\to \mathrm{\infty }}{lim}{y}_n=\sqrt[3]{x}$.

b) Construct a similar algorithm to calculate the fifth root.