Solve the equation \(2 \sin \pi x / 2 - 2 \cos \pi x = x^5 + 10x - 54\).
Find the locus of points whose coordinates \((x, y)\) satisfy the relation \(\sin(x + y) = 0\).
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.