Which of the following inequalities are satisfied for all real numbers a, b, c, x, y, z which satisfy x< a, y< b, and z< c? I. xy+yz+zx< ab+bc+ca II. x^2+y^2+z^2< a^2+b^2+c^2 III. xyz< abc

1975 AMC 12 Problem 3

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Which of the following inequalities are satisfied for all real numbers

a,

b,

c,

x,

y,

z

which satisfy

x\lt a,

y\lt b,

and

z\lt c?

xy+yz+zx\lt ab+bc+ca

II.

x^2+y^2+z^2\lt a^2+b^2+c^2

III.

xyz\lt abc

None are satisfied.

\mathrm{I}

only

\mathrm{II}

only

\mathrm{III}

only

All are satisfied.

The variables are real, so increasing a number need not increase its square or its products

Try

a=b=1,\ c=-1,

x=y=0,\ z=-10

Take

a=b=1,\ c=-1,

and

x=y=0,\ z=-10.

The required three coordinate inequalities hold. But I becomes

0\lt-1,

II becomes

100\lt3,

and III becomes

0\lt-1,

all false. Thus none is universally true.

Therefore, the correct answer is A.