Inequalities, Several

Which of the following inequalities are satisfied for all real numbers a, b, c, x, y, z that satisfy the conditions x < a, y < b, and z < c?

  1. xy + yz + zx < ab + bc + ca
  2. x2 + y2 + z2 < a + b + c
  3. xyz < abc

  1. None is satisfied
  2. I only
  3. II only
  4. III only
  5. All are satisfied


Show/Hide Solution

We search vigorously for counterexamples in which x < a, y < b, and z < c

  1. xy + yz + zx < ab + bc + ca
    Let x = y = z = -1 and a = b = c = 0 and I is false.

  2. x2 + y2 + z2 < a + b + c
    We can use the same numbers to destroy II.

  3. xyz < abc
    Let x = y = -1 and z = 1, and a = b = 0 and c = 2. That wrecks III.

So the answer is (a): None of the inequalities is satisfied.

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