
Problem 10.1. Find the least natural number $a$ such that the equation $\cos ^{2} \pi(a-x)-2 \cos \pi(a-x)+\cos \frac{3 \pi x}{2 a} \cdot \cos \left(\frac{\pi x}{2 a}+\frac{\pi}{3}\right)+2=0$ has a root.


Solution: The roots of the our equations are the common roots of $\cos \pi(x-a)=1$ and $\cos \frac{3 \pi x}{2 a} \cdot \cos \left(\frac{\pi x}{2 a}+\frac{\pi}{3}\right)+1=0$. The roots of the first one are $x=a+2 n, n=0, \pm 1, \pm 2, \ldots$ and the roots of the second one are $x=2 a\left(k-\frac{1}{3}\right), k=0, \pm 1, \ldots$ Therefore $a=\frac{6 n}{6 k-5}$
for some integer numbers $n$ and $k$. It is easy to see now that the least natural number with the required property is $a=6$.
