Difference between revisions of "1957 AHSME Problems/Problem 36"
(Created page with "By AM-GM, we have <cmath>\frac{x+y}{2} \geq \sqrt{xy}</cmath> Substituting, we have <cmath>\frac{1}{2} \geq \sqrt {xy}</cmath> <cmath>\frac{1}{4} \geq xy</cmath> Equality occu...") |
(solution edits, see also box, statement of problem) |
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| − | By AM-GM, we have | + | == Problem == |
| + | If <math>x + y = 1</math>, then the largest value of <math>xy</math> is: | ||
| + | |||
| + | <math>\textbf{(A)}\ 1\qquad \textbf{(B)}\ 0.5\qquad \textbf{(C)}\ \text{an irrational number about }{0.4}\qquad \textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0</math> | ||
| + | |||
| + | == Solution == | ||
| + | By [[AM-GM]], we have | ||
<cmath>\frac{x+y}{2} \geq \sqrt{xy}</cmath> | <cmath>\frac{x+y}{2} \geq \sqrt{xy}</cmath> | ||
Substituting, we have | Substituting, we have | ||
<cmath>\frac{1}{2} \geq \sqrt {xy}</cmath> | <cmath>\frac{1}{2} \geq \sqrt {xy}</cmath> | ||
<cmath>\frac{1}{4} \geq xy</cmath> | <cmath>\frac{1}{4} \geq xy</cmath> | ||
| − | Equality occurs when <math>x = y = \ | + | Equality occurs when <math>x = y = \boxed{\textbf{(D) }\frac12}</math>. |
| − | |||
~JustinLee2017 | ~JustinLee2017 | ||
| + | |||
| + | == See Also == | ||
| + | {{AHSME 50p box|year=1957|num-b=35|num-a=37}} | ||
| + | {{MAA Notice}} | ||
| + | [[Category:AHSME]][[Category:AHSME Problems]] | ||
Latest revision as of 08:13, 26 July 2024
Problem
If 
, then the largest value of 
is:
Solution
By AM-GM, we have
![\[\frac{x+y}{2} \geq \sqrt{xy}\]](http://latex.artofproblemsolving.com/9/9/f/99fa58a672272e2c58e856e035838fac36cebf3c.png)
Substituting, we have
![\[\frac{1}{2} \geq \sqrt {xy}\]](http://latex.artofproblemsolving.com/6/f/0/6f01cae884790a11d14890d3c93664d1ae1d09c1.png)
![\[\frac{1}{4} \geq xy\]](http://latex.artofproblemsolving.com/4/b/6/4b68740b10f0fb21e2e2b88bb702de8e9bc17869.png)
Equality occurs when 
.
~JustinLee2017
See Also
| 1957 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 35 |
Followed by Problem 37 | |
| 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. 
