Difference between revisions of "1977 USAMO Problems/Problem 5"
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== Problem == | == Problem == | ||
If <math> a,b,c,d,e</math> are positive numbers bounded by <math> p</math> and <math> q</math>, i.e, if they lie in <math> [p,q], 0 < p</math>, prove that | If <math> a,b,c,d,e</math> are positive numbers bounded by <math> p</math> and <math> q</math>, i.e, if they lie in <math> [p,q], 0 < p</math>, prove that | ||
| − | <cmath> (a \plus{} b \plus{} c \plus{} d \plus{} e)\left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \plus{} \frac {1}{d} \plus{} \frac {1}{e}\right) \le 25 \plus{} 6\left(\sqrt {\frac {p}{q}} \minus{} \sqrt {\frac {q}{p}}\right)^2</cmath> | + | <cmath> (a \plus{} b \plus{} c \plus{} d \plus{} e)\left(\frac{1}{a} \plus{} \frac {1}{b} \plus{} \frac{1}{c} \plus{} \frac{1}{d} \plus{} \frac{1}{e}\right) \le 25 \plus{} 6\left(\sqrt{\frac {p}{q}} \minus{} \sqrt {\frac{q}{p}}\right)^2</cmath> |
and determine when there is equality. | and determine when there is equality. | ||
Revision as of 14:15, 17 September 2012
Problem
If 
are positive numbers bounded by 
and 
, i.e, if they lie in ![$[p,q], 0 < p$](http://latex.artofproblemsolving.com/b/7/d/b7db0d1c44b17ab521242db4599465126d5ad617.png)
, prove that
\[(a \plus{} b \plus{} c \plus{} d \plus{} e)\left(\frac{1}{a} \plus{} \frac {1}{b} \plus{} \frac{1}{c} \plus{} \frac{1}{d} \plus{} \frac{1}{e}\right) \le 25 \plus{} 6\left(\sqrt{\frac {p}{q}} \minus{} \sqrt {\frac{q}{p}}\right)^2\] (Error compiling LaTeX. Unknown error_msg)
and determine when there is equality.
Solution
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See Also
| 1977 USAMO (Problems • Resources) | ||
| Preceded by Problem 4 |
Followed by Last Question | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
