Problem
ALG-B2-M04-P005 Squares versus mixed products
#5
★★☆☆☆ Level 2 of 5
Prove for \(a,b,c\ge0\): \[a^2+b^2+c^2\ge ab+bc+ca.\]
Hint. Order the numbers and compare the correctly paired product sum with a cyclic permutation.
Order \(a,b,c\) as \(u_1\le u_2\le u_3\). By rearrangement, \(u_1^2+u_2^2+u_3^2\) is the largest among sums \(\sum u_i u_{\sigma(i)}\). The sum \(ab+bc+ca\) is one of these permutations, so it is not larger than the sum of squares.
The problem can also be solved by squared differences, but here the point is to see its rearrangement structure.