7.4.1. (12 points) How many integer roots of the equation

$$
\cos 2 \pi x + \cos \pi x = \sin 3 \pi x + \sin \pi x
$$

lie between the roots of the equation $x^{2} + 10 x - 17 = 0$?
# Answer: 7.

Solution. If $x=2 k, k \in \mathbb{Z}$, then $\cos 2 \pi x+\cos \pi x=\cos 4 \pi k+\cos 2 \pi k=2, \sin 3 \pi x+\sin \pi x=$ $\sin 6 \pi k+\sin 2 \pi k=0$, so there are no even numbers among the roots of the first equation.

If, however, $x=2 k+1, k \in \mathbb{Z}$, then $\cos 2 \pi x+\cos \pi x=\cos (4 \pi k+2 \pi)+\cos (2 \pi k+\pi)=0, \sin 3 \pi x+\sin \pi x=$ $\sin (6 \pi k+3 \pi)+\sin (2 \pi k+\pi)=0$, so all odd numbers are roots of the first equation.

The roots of the second equation are $-5 \pm \sqrt{42}$; the smaller one lies between -12 and -11, and the larger one lies between 1 and 2. Thus, between the roots of the second equation lie 7 odd numbers $-11,-9,-7,-5,-3,-1,1$, which are 7 integer roots of the first equation.