Given your parameters there should be profit. There can be profit even in perfect competition if there is less than infinite firms since as pointed out by Bayesian in his +1 comment when price is equal marginal cost there is no profit only on the last unit sold. Here is the full explanation:
The profit function of a firm is given by:
$$\pi = pq_i - aq_i^2$$
So FOC of this problem is given by:
$$\frac{d \pi }{dq_i}= p -2aq_i $$
and hence in the optimum:
$$\frac{p}{2a} = q_i^* $$
Now you can plug the optimum solution above back in to profit function and:
$$\pi = p\frac{p}{2a} - a\left(\frac{p}{2a}\right)^2 = \frac{p^2}{2a} - \frac{p^2}{4a} = \frac{p^2}{4a} = \frac{p^2}{8} \text{ for } a=2$$
Also the supply is given by the sum of optimum quantities across the whole market so:
$$S = \sum^n q_i^* = n(\frac{p}{2a}) = 45p \text{ for } a=2 \text{ and } n =180 $$
where the above assumes all firms are same so that $\sum^n q_i^* = nq^*$
Now equilibrium price will be given by intersection of supply and demand:
$$100−5p = n(\frac{p}{2a}) \implies p^*= \frac{200a}{n+10a}$$
Now the last expression goes to zero as the number of firms increases. However, for $n=180$ and $a=2$ we have:
$$p^*= \frac{400}{200}=2 \implies Q* = \sum^n q_i^* = 90$$
Meaning every firm will produce $q_i^* = 90/180 = 1/2$ and that the profits will be: $1/2$.
However, note the above can be sustained only if you assume no new firms can enter the market. In the long run with free entry - new firms will enter the market up until the $n$ is such that economic profit is zero (remember economic profit is not accounting profit so even at $\pi=0$ people have incentive to be doing business).