9.1 Karlson eats three jars of jam and one jar of honey in 25 minutes, while Little Man takes 55 minutes. One jar of jam and three jars of honey Karlson eats in 35 minutes, while Little Man takes 1 hour 25 minutes. How long will it take them to eat six jars of jam together?
9.1 20 minutes.

From the condition, it follows that if Karlson eats three jars of jam and one jar of honey, and then immediately eats one jar of jam and three jars of honey, he will spend $25+35=60$ minutes. For Little Man, this time will be 140 minutes. Therefore, Karlson will spend 15 minutes on one jar of jam and one jar of honey, and 10 minutes on two jars of jam, which is $25-15=10$ minutes. For Little Man, this time is 35 minutes and 20 minutes, respectively. In the end, they will eat 6 jars in 20 minutes.

9.2 $a=b=c=1$.

Let's move all the terms to the left side. After transformations, we get $a(a-1)^{2}+b(b-1)^{2}+c(c-1)^{2}=0$. Since $a, b, c$ are positive numbers, on the left we have a sum of non-negative terms. Their sum is zero only if each term is zero. This is possible only when $a=b=c=1$.