5. Rewrite the equation as

$$
5^{(x-1)^{2}} \log _{7}\left((x-1)^{2}+2\right)=5^{2|x-a|} \log _{7}(2|x-a|+2)
$$

Let $f(t)=5^{t} \log _{7}(t+2)$. This function is increasing. Therefore,

$$
f\left((x-1)^{2}\right)=f(2|x-a|) \Longleftrightarrow(x-1)^{2}=2|x-a|
$$

There will be three solutions in the case of tangency for $a=1 / 2, a=3 / 2$ and when $a=1$, since the vertices of the parabola $y=(x-1)^{2}$ and the curve $y=|x-1|$ coincide.
Answer: $a=1 / 2 ; 1 ; 3 / 2$.

Answer to the option: $6-2: a=-3 / 2 ;-1 ;-1 / 2$.

## Lomonosov Moscow State University
## Olympiad "Conquer Sparrow Hills" Variant 7-1