Cournot Competition game with 3 Firms
Firm 1's profit maximization problem is :
\begin{eqnarray*} \max_{q_1} &amp; \ p(q_1 + q_2 + q_3)q_1 - c(q_1) \end{eqnarray*}
Firm 1's response to firm 2's and 3's quantity choices will satisfy the following:
\begin{eqnarray*} p(q_1 + q_2 + q_3) + p'(q_1 + q_2 + q_3)q_1 - c'(q_1) = 0 \end{eqnarray*}
Likewise, we'll get the implicit best response functions for the other firms as well. 
\begin{eqnarray*} p(q_1 + q_2 + q_3) + p'(q_1 + q_2 + q_3)q_i - c'(q_i) = 0 \end{eqnarray*}
for each $i \in \{1,2 ,3\}$.

There is a symmetric equilibrium of the above game if there exist $q^*$ such that
\begin{eqnarray*} p(3q^*) + p'(3q^*)q^* - c'(q^*) = 0 \end{eqnarray*}
To show that there exist one, consider the following function
\begin{eqnarray*} g(q) = p(3q) + p'(3q)q - c'(q)\end{eqnarray*}
Observe that $g(0) = p(0) - c'(0) &gt; 0$. Also, given that $p(q)$ is decreasing and concave, there exist $\overline{q} &gt; 0$ such that $p(\overline{q}) \leq 0$. Consequently, $g(\overline{q}) \leq 0$. Therefore, by intermediate value theorem, there exist $q^* \in [0,\overline{q}] $ such that $g(q^*) = 0$ i.e.
\begin{eqnarray*} p(3q^*) + p'(3q^*)q^* - c'(q^*) = 0 \end{eqnarray*}
Also, $g'(q) = 4p'(3q) + 3p''(3q)q - c''(q) &lt; 0$. Since $g(q)$ is decreasing in $q$, $q^*$ is unique.
So there is a unique symmetric equilibrium. To show that it is the only equilibrium, we'll show that there is no equilibrium where at least two of the firms choose different quantities. To do so, let us subtract the implicit best response function of firm 1 from that of firm 2, 
\begin{eqnarray*} p'(q_1 + q_2 + q_3)(q_2 - q_1) = c'(q_2) - c'(q_1) \end{eqnarray*}
Notice that the above equality only holds for $q_2 = q_1$. This is because 


when $q_2 &gt; q_1$, $p'(q_1 + q_2 + q_3)(q_2 - q_1) &lt; 0$ and $c'(q_2) - c'(q_1) &gt; 0$, 
when $q_2 &lt; q_1$, $p'(q_1 + q_2 + q_3)(q_2 - q_1) &gt; 0$ and $c'(q_2) - c'(q_1) &lt; 0$.


Therefore, there is no equilibrium satisfying $q_2 \neq q_1$. Likewise, there is none satisfying $q_2 \neq q_3$ or $q_1 \neq q_3$.