AHSME 1951 Problem 43

Of the following statements, the only one that is incorrect is: $\textbf{(A)}\ \text{An inequality will remain true after each side is increased,}$ $\text{ decreased, multiplied or divided zero excluded by the same positive quantity.}$ $\textbf{(B)}\ \text{The arithmetic mean of two unequal positive quantities is greater than their geometric mean.}$ $\textbf{(C)}\ \text{If the sum of two positive quantities is given, ther product is largest when they are equal.}$ $\textbf{(D)}\ \text{If }a\text{ and }b\text{ are positive and unequal, }\frac{1}{2}(a^{2}+b^{2})\text{ is greater than }[\frac{1}{2}(a+b)]^{2}.$ $\textbf{(E)}\ \text{If the product of two positive quantities is given, their sum is greatest when they are equal.}$