this question is interesting. the direction is arithmetic mean is always no less then geometry mean. they equal if all positive nums are equal. this is a well-known inequality, called cauchy inequality if im not wrong.

my hint is as follow (using mi)

for 2 numbers: (a+b)/2 >= sqrt(a+b)

assume that (a1+a2+....+an)/n >= sqrt[n](a1xa2x....xan)

prove that the inequality still hold for n+1

during the proof, we can easily see that "=" happens when all num are equal.