In the centre of a rectangular billiard table that is 3 m long and 1 m wide, there is a billiard ball. It is hit by a cue in a random direction. After the impact the ball stops passing exactly 2 m. Find the expected number of reflections from the sides of the table.
Solve the equation \(2 \sin \pi x / 2 - 2 \cos \pi x = x^5 + 10x - 54\).
The number \(x\) is such that both the sums \(S = \sin 64x + \sin 65x\) and \(C = \cos 64x + \cos 65x\) are rational numbers.
Prove that in both of these sums, both terms are rational.
Find the locus of points whose coordinates \((x, y)\) satisfy the relation \(\sin(x + y) = 0\).
Prove that the sequence \(x_n = \sin (n^2)\) does not tend to zero for \(n\) that tends to infinity.
Does there exist a flat quadrilateral in which the tangents of all interior angles are equal?
The numbers \(a\) and \(b\) are such that the first equation of the system \[\begin{aligned} \sin x + a & = bx \\ \cos x &= b \end{aligned}\] has exactly two solutions. Prove that the system has at least one solution.
The numbers \(a\) and \(b\) are such that the first equation of the system \[\begin{aligned} \cos x &= ax + b \\ \sin x + a &= 0 \end{aligned}\] has exactly two solutions. Prove that the system has at least one solution.