Difference between revisions of "1953 AHSME Problems/Problem 45"
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| − | + | ==Problem== | |
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| + | The lengths of two line segments are <math>a</math> units and <math>b</math> units respectively. Then the correct relation between them is: | ||
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| + | <math>\textbf{(A)}\ \frac{a+b}{2} > \sqrt{ab} \qquad | ||
| + | \textbf{(B)}\ \frac{a+b}{2} < \sqrt{ab} \qquad | ||
| + | \textbf{(C)}\ \frac{a+b}{2}=\sqrt{ab}\\ \textbf{(D)}\ \frac{a+b}{2}\leq\sqrt{ab}\qquad | ||
| + | \textbf{(E)}\ \frac{a+b}{2}\geq\sqrt{ab} </math> | ||
| + | |||
| + | ==Solution== | ||
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| + | Since both lengths are positive, the [[AM-GM Inequality]] is satisfied. The correct relationship between <math>a</math> and <math>b</math> is <math>\boxed{\textbf{(E)}\ \frac{a+b}{2}\geq\sqrt{ab}}</math>. | ||
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| + | ==See Also== | ||
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| + | {{AHSME 50p box|year=1953|num-b=44|num-a=46}} | ||
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| + | {{MAA Notice}} | ||
Latest revision as of 23:02, 14 February 2020
Problem
The lengths of two line segments are 
units and 
units respectively. Then the correct relation between them is:
Solution
Since both lengths are positive, the AM-GM Inequality is satisfied. The correct relationship between and is 
.
See Also
| 1953 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 44 |
Followed by Problem 46 | |
| 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. 
