I'd go to Rosetta Code and pick ten tasks that looks to play into that language's strength.
But there are two problems with the question, as stated. First, some languages play to some strengths and some to others, so no set of ten problems will ever be a good exercise for a language.
But there's a even more serious problem with this, even assuming that problem is overcome. Once you did a problem in one language -- in fact, once you did a problem in three or four languages that share a similar style, you'll tend to approach that problem the same way with every new language, which is exceedingly counter-productive when trying to learn a new language.
In my opinion, doing familiar problems is a good way to learn the new syntax of things you already know in a new language. In a sense, it is a good fast forward, even if it often leaves holes in one's knowledge that one won't even notice are there.
But to truly know what makes a language different, what pushed someone to create a new language, you have to approach it with a problem you don't know how to solve.