Cost function and short run supply
Just like a Production function, in Economics, a Cost function is not an "engineering relation", but a "best response", reflecting minimum cost to produce a given quantity $q$.  

Cost-minimizing firms are at first assumed not to determine their scale of operations -what they decide upon is the demand for production inputs,given the inputs' prices.  

Still, by duality, the profit-maximizing condition should also hold -these are for-profit private companies, not, say, some public utility without a profit motive. If the firms are price-takers in the diamonds market, this relation is well-known, and provides the answer to a) (remember that there are functions, and then there are inverse functions).  

As for b), I see in the question that the suppliers are assumed identical.