Question 3
Let $P_{i}(x) = x^{2} + b_{i} x + c_{i}$ for $i = 1, 2, \cdots, n$ be pairwise distinct polynomials of degree $2$ with real coefficients so that for any $0 \leq i \lt j \leq n$; $i, j \in \mathbb{N}$, the polynomial $Q_{i, j}(x) = P_{i}(x) + P_{j}(x)$ has only one real root. Find the greatest possible value of $n$.