Price and quantity in equilibrium I have a market with 1000 suppliers.

The total demand can be described by

$$
Q=-300000 P + 6000000
$$

while the total supply can be described by

$$
Q=200000 \sum(MC^2)-200000
$$

What is the price and quantity in equilibrium?

I have no idea what to do with $\sum(MC^2)$ in the supply.

I guess I should isolate $P$ in the first equation:
$$
P(Q) = 20-\frac{1}{300000} Q
$$
but I don't know what to do now.
Hint: To find equilibrium price and quantity, you want to find when the system is in equilibrium. In equilibrium, quantity demanded should equal quantity supplied.

$$Q_d = Q_s \implies -300,000P + 6,000,000 = 200,000 \sum_{i=1}^n (c_i'(q_i))^2 - 200,000$$

Where $c_i(q_i)$ is the cost function for firm $i$ with respect to $q_i$, firm $i$'s production. I'm assuming when you wrote $MC$ you meant marginal cost.

$$\implies 6,200,000 - 200,000\sum_{i=1}^n (c_i'(q_i))^2 = 300,000P$$

$$\implies P = \frac{62}{3} - \frac{2}{3} \sum_{i=1}^n (c_i'(q_i))^2$$

Now that we have an inverse demand function with respect to $q$, we have to solve the firms' maximization problem. Since there's no information you've given to us directly about that, let's assume technology is homogeneous and that firms compete via Cournot competition.

$$\Pi_j = q_j(P(q)) - c(q_j)$$

$$\Pi_j =  q_j(\frac{62}{3} - \frac{2}{3} \sum_{i=1}^n (c'(q_i))^2) - c(q_j)$$

Now take the derivative w.r.t $q_j$ and set to zero to find the maximum.

$$\Pi_j' =  (\frac{62}{3} - \frac{2}{3} \sum_{i=1, i \neq j}^n (c'(q_i))^2) + \frac{2}{3} \frac{d}{dq_j} (c'(q_j)^2 \cdot q_j) - c'(q_j) = 0$$

Where that derivative is evaluated with the product and chain rule.

From here, you solve for $q_j$ as an expression of all $q_{i, i \neq j}$. Since the cost functions are homogeneous, it shouldn't be that bad to do. Also remember that the number of suppliers is 1000, so the problem can be further simplified:

$$\Pi_j' =  (\frac{62}{3} - \frac{2}{3} \cdot 999(c'(q_i))^2) + \frac{2}{3} \frac{d}{dq_j} (c'(q_j)^2 \cdot q_j) - c'(q_j) = 0$$

$$\implies \Pi_j' =  (\frac{62}{3} - 666(c'(q_i))^2) + \frac{2}{3} (2c'(q_j) \cdot c''(q_j) \cdot q_j + c'(q_j)^2) - c'(q_j) = 0$$

and then you solve for $q_j$ in terms of $q_{i, i \neq j}$ and do the rest of the Cournot derivation.