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?
There are 4 weights and scales. How many loads that are different by weight can be accurately weighed using these weights, if
a) weights can be placed only on one side of the scales;
b) weights can be placed on both sides of the scales?
Will thought of a number: 1, 2 or 3. You can ask him only one question, to which he can answer “yes”, “no” or “I do not know”. Can you guess the number by asking just one question?
Is it possible to draw from some point on a plane \(n\) tangents to a polynomial of \(n\)-th power?
Prove the following formulae are true: \[\begin{aligned} a^{n + 1} - b^{n + 1} &= (a - b) (a^n + a^{n-1}b + \dots + b^n);\\ a^{2n + 1} + b^{2n + 1} &= (a + b) (a^{2n} - a^{2n-1}b + a^{2n-2}b^2 - \dots + b^{2n}). \end{aligned}\]