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.
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 \((\sqrt 2, \sqrt 3)\).
Is it possible to draw from some point on a plane \(n\) tangents to a polynomial of \(n\)-th power?
Prove that if \((p, q) = 1\) and \(p/q\) is a rational root of the polynomial \(P (x) = a_nx^n + \dots + a_1x + a_0\) with integer coefficients, then
a) \(a_0\) is divisible by \(p\);
b) \(a_n\) is divisible by \(q\).
Prove that the root a of the polynomial \(P (x)\) has multiplicity greater than 1 if and only if \(P (a) = 0\) and \(P '(a) = 0\).
Prove that the following polynomial does not have any identical roots: \(P(x) = 1 + x + x^2/2! + \dots + x^n/n!\)
For which \(A\) and \(B\) does the polynomial \(Ax^{n + 1} + Bx^n + 1\) have the number \(x = 1\) at least two times as its root?