1951 AMC 12 Problem 43

Question

Of the following statements, the only one that is incorrect is:

An inequality will remain true after each side is increased, decreased, multiplied, or divided (zero excluded) by the same positive quantity.

The arithmetic mean of two unequal positive quantities is greater than their geometric mean.

If the sum of two positive quantities is given, their product is largest when they are equal.

If

a

and

b

are positive and unequal,

\dfrac12(a^2+b^2)

is greater than

\left[\dfrac12(a+b)\right]^2.

If the product of two positive quantities is given, their sum is greatest when they are equal.

For a fixed positive product, compare the sum at equality with sums from increasingly unequal factors

AM-GM gives a minimum, not a maximum, for the sum when the product is fixed

If

uv=P\gt0,

then

u+v\ge2\sqrt P,

with equality at

u=v.

Thus equality gives the least possible sum. The sum can grow without bound by taking

u

large and

v=\frac{P}{u}

small, so it is not greatest at equality.

Thus, the correct answer is E.

Problem 43 in other years