I found this interesting (to me, at least) when I was solving this problem:
"Let \(a_1,a_2,...,a_n,b,k\) be positive reals so that \(a_1+a_2+...+a_n \le nk\). Prove that: \((a_1+b)(a_2+b)...(a_n+b) \le (k+b)^n \)."
At first, my thought was: "Oh, so it's obvious that \(k^n \ge a_1*a_2* ... *a_n\). But what about the \(b\)? There's no way to throw it away." And right now, I'm stuck at that exact point. Reverse Rearrangement doesn't seem to work, and there is a "seemingly" trivial solution using Buffalo Way, but I want to find a "better" way than just plug-and-bash.
So, my question is: Is there a "good-looking" way to solve the problem? If there is, what is your inspiration?
Any help would be greatly appreciated. Thanks for reading!
P/S: If you can find any counterexamples, please tell me immediately and I will delete/edit this note.