2210.51 – In-equali-tease

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

  1. xy + yz + zx < ab + bc + caxy + yz + zx < ab + bc + ca

  2. x2 + y2 + z2 < a + b + cx^2 + y^2 + z^2 < a + b + c

  3. xyz < abcxyz < abc

  4. None is satisfied

  5. I only

  6. II only

  7. III only

  8. All are satisfied.

Solution

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

I. xy + yz + zx < ab + bc + ca . Let x = y = z =−1x = y = z = -1 and a = b = c =0a = b = c = 0, and I is done, i.e., this inequality is false.

II. x2 + y2 + z2 < a + b + cx^2 + y^2 + z^2 < a + b + c. The same numbers disprove this one.

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

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