Difference between revisions of "1951 AHSME Problems/Problem 43"
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| + | ==Problem== | ||
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| + | Of the following statements, the only one that is incorrect is: | ||
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| + | <math> \textbf{(A)}\ \text{An inequality will remain true after each side is increased,} </math> <math>\text{ decreased, multiplied or divided zero excluded by the same positive quantity.} </math> | ||
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| + | <math> \textbf{(B)}\ \text{The arithmetic mean of two unequal positive quantities is greater than their geometric mean.} </math> | ||
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| + | <math> \textbf{(C)}\ \text{If the sum of two positive quantities is given, ther product is largest when they are equal.} </math> | ||
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| + | <math> \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}. </math> | ||
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| + | <math> \textbf{(E)}\ \text{If the product of two positive quantities is given, their sum is greatest when they are equal.} </math> | ||
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| + | ==Solution== | ||
| + | The answer is <math>\boxed{\textbf{(E)}}</math>. | ||
| + | Quite the opposite of statement (E) is true--the sum <math>a+b</math> is <i>minimized</i> when <math>a=b</math>, but it approaches <math>\infty</math> when one of <math>a,b</math> gets arbitrarily small. | ||
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== See Also == | == See Also == | ||
{{AHSME 50p box|year=1951|num-b=42|num-a=44}} | {{AHSME 50p box|year=1951|num-b=42|num-a=44}} | ||
Latest revision as of 10:01, 19 April 2014
Problem
Of the following statements, the only one that is incorrect is:
![$\textbf{(D)}\ \text{If }a\text{ and }b\text{ are positive and unequal, }\frac{1}{2}(a^{2}+b^{2})\text{ is greater than }[\frac{1}{2}(a+b)]^{2}.$](http://latex.artofproblemsolving.com/0/a/b/0ab4e2d144ac92041cb049e6899919de6beecd2b.png)
Solution
The answer is 
.
Quite the opposite of statement (E) is true--the sum 
is minimized when 
, but it approaches 
when one of 
gets arbitrarily small.
See Also
| 1951 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 42 |
Followed by Problem 44 | |
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| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
