1951 AHSME Problems/Problem 43

Revision as of 11:01, 19 April 2014 by Hukilau17 (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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 (ProblemsAnswer KeyResources)
Preceded by
Problem 42
Followed by
Problem 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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Invalid username
Login to AoPS