1951 AHSME 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, their 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.}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.