Problems

Prove the irrationality of the following numbers:

a) $\sqrt{3}17$

b) $\sqrt{2}+\sqrt{3}$

c) $\sqrt{2}+\sqrt{3}+\sqrt{5}$

d) $\sqrt{3}3-\sqrt{2}$

e) $\cos {10}^{\circ }$

f) $\tan {10}^{\circ }$

g) $\sin 1^{\circ }$

h) ${\log }_{2}3$

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?

One of the roots of the equation $x^2+ax+b=0$ is $1+\sqrt{3}$. Find $a$ and $b$ if you know that they are rational.

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

For what natural numbers $a$ and $b$ is the number ${\log }_{a}b$ rational?

Prove that for $x\ne \pi n$ ($n$ is an integer) $\sin x$ and $\cos x$ are rational if and only if the number $\tan x/2$ is rational.

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 there is at most one point of an integer lattice on a circle with centre at $(\sqrt{2},\sqrt{3})$.

$N$ points are given, no three of which lie on one line. Each two of these points are connected by a segment, and each segment is coloured in one of the $k$ colours. Prove that if $N>\lfloor k!e\rfloor$, then among these points one can choose three such that all sides of the triangle formed by them will be colored in one colour.

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?