Question 70
We call an integer number $n \gt 0$ interesting if for each permutation $\sigma$ of $1, 2, \dots, n$ there exist polynomials $P_1, P_2, \dots, P_n$ and $\epsilon \gt 0$ such that:
$$
i)\ P_1(0) = P_2(0) = \dots = P_n(0).
$$
$$
ii)\ P_1(x) \gt P_2(x) \gt \dots \gt P_n(x) \text{ for } -\epsilon \lt x \lt 0.
$$
$$
iii)\ P_{\sigma(1)}(x) \gt P_{\sigma(2)}(x) \gt \dots \gt P_{\sigma(n)}(x) \text{ for } 0 \lt x \lt \epsilon.
$$
Find all interesting numbers.
$$
i)\ P_1(0) = P_2(0) = \dots = P_n(0).
$$
$$
ii)\ P_1(x) \gt P_2(x) \gt \dots \gt P_n(x) \text{ for } -\epsilon \lt x \lt 0.
$$
$$
iii)\ P_{\sigma(1)}(x) \gt P_{\sigma(2)}(x) \gt \dots \gt P_{\sigma(n)}(x) \text{ for } 0 \lt x \lt \epsilon.
$$
Find all interesting numbers.