Problem
ALG-B2-M12-P016 Set 4. Telescoping root
#16
★★★★★ Level 5 of 5
Let \(x_1,x_2,x_3,x_4>0\), \(x_5=x_1\). Prove \[\sum_{i=1}^4(x_{i+1}-x_i)\sqrt{x_i^2+3x_{i+1}^2}\ge0.\]
Hint. Compare the root with \(x_i+x_{i+1}\), keeping track of the sign of \(x_{i+1}-x_i\).
For \(u,v>0\), \((v-u)\sqrt{u^2+3v^2}\ge v^2-u^2\): if \(v\ge u\), the root is at least \(u+v\); if \(v\le u\), the root is at most \(u+v\), but the multiplier is negative. The right-hand sides telescope to \(0\).
Mock set 4: the problem is intended for independent method selection without an explicit cue in the statement.