62nd Ukrainian National Mathematical Olympiad, Third Round, Second Tour

If numbers $a$, $b$, $c$ have a common divisor, we can divide by it and get a triple of integers $a_1$, $b_1$, $c_1$, whose largest common divisor is $1$. As $a_1+(a_1,b_1)=b_1+(b_1,c_1)$, $(b_1,c_1)$ is divisible by $(a_1,b_1)$. But $((a_1,b_1),(b_1,c_1))=1$, as explained above, so $(a_1,b_1)=1$. Similarly we get that our numbers are pairwise coprime, so $a_1+1=b_1+1=c_1+1$, implying $a=b=c$.