
Problem 1. Let $A=\frac{1}{\sqrt{4 x^{2}+4 x+1}}$ and $B=\frac{2 x-2}{\sqrt{x^{2}-2 x+1}}$. Find all integer values of $x$, for which the number $C=\frac{2 A+B}{3}$ is an integer.


Solution. We have

$$
A=\frac{1}{\sqrt{(2 x+1)^{2}}}=\frac{1}{|2 x+1|}, \quad B=\frac{2(x-1)}{\sqrt{(x-1)^{2}}}=\frac{2(x-1)}{|x-1|}
$$

and

$$
C=\frac{2}{3} \cdot\left(\frac{1}{|2 x+1|}+\frac{x-1}{|x-1|}\right)
$$

1. Let $x>1$. Then

$$
C=\frac{2}{3} \cdot\left(\frac{1}{2 x+1}+1\right)=\frac{4(x+1)}{3(2 x+1)}>0
$$

and

$$
C-1=\frac{4(x+1)}{3(2 x+1)}-1=\frac{1-2 x}{3(2 x+1)}1$.

2. Let $-\frac{1}{2}0
$$

Hence $-1<C \leq 0$, i.e. $C=0$ and $x=-1$.

Finally only $x=0$ and $x=-1$ are solutions of the given problem.
