Question 1
Prove that the following statement is true for $n=3$ and for $n=5$, and false for all other $n\gt 2$:
For any real numbers $a_1,a_2,\ldots,a_n$,
$$(a_1-a_2)(a_1-a_3)\cdots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\cdots(a_2-a_n)+\ldots$$
$$+(a_n-a_1)(a_n-a_2)\cdots(a_n-a_{n-1})\ge0.$$
For any real numbers $a_1,a_2,\ldots,a_n$,
$$(a_1-a_2)(a_1-a_3)\cdots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\cdots(a_2-a_n)+\ldots$$
$$+(a_n-a_1)(a_n-a_2)\cdots(a_n-a_{n-1})\ge0.$$