$P(Q)$ represents a market where demand $Q$ is related to price $P$ by $$P(Q) = Q^{-\frac{1}{2}}$$ In this market there are $m$ identical producers, say firm 1, 2, up to $m$ which can produce any non-negative quantity say $q$ at costs $$q^2$$
We make the profit function: $$\Pi=Q^{-\frac{1}{2}}\cdot q -q^2$$ For firm $q_i$: $$\Pi=q_i\cdot\sum q_{-i} - q_i^2$$ Deriving and set to $0$: $$q_i=\frac{2m-1}{4m}$$ And $$q*=\frac{1}{m-1}$$
Is this ok? I just don't know anymore, have been working for hours on this......