# 1951 AHSME Problems/Problem 43

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## Problem

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

$\textbf{(A)}\ \text{An inequality will remain true after each side is increased,}$ $\text{ decreased, multiplied or divided zero excluded by the same positive quantity.}$

$\textbf{(B)}\ \text{The arithmetic mean of two unequal positive quantities is greater than their geometric mean.}$

$\textbf{(C)}\ \text{If the sum of two positive quantities is given, ther product is largest when they are equal.}$

$\textbf{(D)}\ \text{If }a\text{ and }b\text{ are positive and unequal, }\frac{1}{2}(a^{2}+b^{2})\text{ is greater than }[\frac{1}{2}(a+b)]^{2}.$

$\textbf{(E)}\ \text{If the product of two positive quantities is given, their sum is greatest when they are equal.}$

## Solution

The answer is $\boxed{\textbf{(E)}}$. Quite the opposite of statement (E) is true--the sum $a+b$ is minimized when $a=b$, but it approaches $\infty$ when one of $a,b$ gets arbitrarily small.

## See Also

 1951 AHSC (Problems • Answer Key • Resources) Preceded byProblem 42 Followed byProblem 44 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 All AHSME Problems and Solutions

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