Problem
ALG-B2-M12-P013 Set 4. Two AM-GM estimates
#13
★★★★☆ Level 4 of 5
Let \(a,b,c>0\), \(abc=1\). Prove \[a+b+c+ab+bc+ca\ge6.\]
Hint. Estimate the two sums separately.
By AM-GM, \(a+b+c\ge3\). Also \(ab+bc+ca\ge3\sqrt[3]{a^2b^2c^2}=3\). Add the estimates.
Mock set 4: the problem is intended for independent method selection without an explicit cue in the statement.