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?
 xy + yz + zx < ab + bc + ca
 x^{2} + y^{2} + z^{2} < a + b + c
 xyz < abc
 None is satisfied
 I only
 II only
 III only
 All are satisfied
We search vigorously for counterexamples in which x < a, y < b, and z < c

xy + yz + zx < ab + bc + ca
Let x = y = z = 1 and a = b = c = 0 and I is false. 
x^{2} + y^{2} + z^{2} < a + b + c
We can use the same numbers to destroy II. 
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.