Prove that for a real positive \(\alpha\) and a positive integer \(d\), \(\lfloor \alpha / d\rfloor = \lfloor \lfloor \alpha\rfloor / d\rfloor\) is always satisfied.
Prove that if \(p\) is a prime number and \(1 \leq k \leq p - 1\), then \(\binom{p}{k}\) is divisible by \(p\).
Prove that if \(p\) is a prime number, then \((a + b)^p - a^p - b^p\) is divisible by \(p\) for any integers \(a\) and \(b\).
Prove that there are infinitely many composite numbers among the numbers \(\lfloor 2^k \sqrt{2}\rfloor\) (\(k = 0, 1, \dots\)).
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.
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?