Problems

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 $$\begin{array}{rl}\sin x+a& =bx\\ \cos x& =b\end{array}$$ 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 $$\begin{array}{rl}\cos x& =ax+b\\ \sin x+a& =0\end{array}$$ has exactly two solutions. Prove that the system has at least one solution.