Question 172
Find the smallest positive integer $n$ such that there exist $n$ real numbers $a_1, a_2, \dots, a_n$ satisfying the following conditions:
i) $a_1 + \dots + a_n \gt 0$;
ii) $a_1^3 + \dots + a_n^3 \lt 0$; and
iii) $a_1^5 + \dots + a_n^5 \gt 0$.
i) $a_1 + \dots + a_n \gt 0$;
ii) $a_1^3 + \dots + a_n^3 \lt 0$; and
iii) $a_1^5 + \dots + a_n^5 \gt 0$.