Question 154
For an integer $n \gt 1$, consider $n$ cubic polynomials $P_{1}(x), \ldots, P_{n}(x)$ such that each polynomial has 3 distinct real roots. Denote by $S$ the set of roots of the following equation
$$
P_{1}(x) P_{2}(x) P_{3}(x) \ldots P_{n}(x) = 0.
$$
It is also known that for each $1 \leq i \lt j \leq n$, $P_{i}(x) P_{j}(x) = 0$ has 5 distinct real roots.
1. Prove that if for each $a, b \in S$, there is exactly one $i \in \{1,2,3, \ldots, n\}$ such that $P_{i}(a) = P_{i}(b) = 0$, then $n = 7$.
2. Prove that if $n \gt 7$ then $|S| = 2n + 1$.
$$
P_{1}(x) P_{2}(x) P_{3}(x) \ldots P_{n}(x) = 0.
$$
It is also known that for each $1 \leq i \lt j \leq n$, $P_{i}(x) P_{j}(x) = 0$ has 5 distinct real roots.
1. Prove that if for each $a, b \in S$, there is exactly one $i \in \{1,2,3, \ldots, n\}$ such that $P_{i}(a) = P_{i}(b) = 0$, then $n = 7$.
2. Prove that if $n \gt 7$ then $|S| = 2n + 1$.