Problem
ALG-B2-M04-P016 Four-term cycle
#16
★★★★☆ Level 4 of 5
Prove for \(a,b,c,d>0\): \[a^4+b^4+c^4+d^4\ge a^3b+b^3c+c^3d+d^3a.\]
Hint. Order the numbers as \(u_1\le u_2\le u_3\le u_4\). The right side becomes some permutation of products \(u_i^3\cdot u_j\).
After ordering, the sequences \(u_i^3\) and \(u_i\) have the same order. By rearrangement, the largest product sum \(u_i^3\cdot u_{\sigma(i)}\) is \(\sum u_i^4\). The cyclic right side is one of the permutations, so it is not larger than \(\sum u_i^4\).
This extends the three-variable idea to four variables.