I'm a high school student, and I'm stuck on part 2 of this question. I would like some hints (not a full solution) on how to approach it:
The specific part of the question that confuses me is the significance of [Hint: The case $n=4$ is a good place to start.]. What about case $n=4$ makes it better over case $n=3$? Because I'm failing to see that.
What I've tried so far with case $n=4$ is that I've tried to show $((a+b+c+d)/4)^4 \geq abcd$ by expanding $(a+b+c+d)/4$. I don't think this is the right method though, for two reasons:
It doesn't seem to use the fact that $n=4$: I could have done $n=3$, with $((a+b+c)/3)^3 \geq abc$.
I don't see how I can generalize this method to an AM-GM inequality with any $n$ values.
So could someone give me a hint about the hint :)? I would really appreciate it if someone could explain how I could use this hint to solve this problem. Also, could you give me a hint only, and not a full solution? I still want solving the remaining problem to be a challenge.
Thanks in advance!