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?

I. xy+yz+zx<ab+bc+caxy + yz + zx < ab + bc + ca
II. x2+y2+z2<a+b+cx^2 + y^2 + z^2 < a + b + c
III. xyz<abcxyz < abc

  1. None is satisfied

  2. I only

  3. II only

  4. III only

  5. 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+caxy + 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=−1x = y = -1 and z=1z = 1, and a=b=a = b = 0 and c=2c = 2. That obliterates III.

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