Problem 10.1. Solve the equation

$$
(x+6) 5^{1-|x-1|}-x=(x+1)\left|5^{x}-1\right|+5^{x+1}+1
$$

Ivan Landjev
10.1. For $x \geq 1$ the equation becomes $(x+6)\left(5^{x}-5^{2-x}\right)=0$, whence $x=2-x$, т.е. $x=1$. For $0 \leq x<1$ the equation becomes an identity, i.e. every $x \in[0,1)$ is a solution. For $x<0$ we obtain $2\left(5^{x}-1\right)(x+1)=0$, whence $x=-1$.

The solutions of the problem are $x \in[0,1] \cup\{-1\}$.