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 | + | <cmath> (a+b +c +d +e)\left(\frac{1}{a} +\frac {1}{b} +\frac{1}{c} + \frac{1}{d} +\frac{1}{e}\right) \le 25 + 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 15:16, 17 September 2012
Problem
If are positive numbers bounded by
and
, i.e, if they lie in
, prove that
\[(a+b +c +d +e)\left(\frac{1}{a} +\frac {1}{b} +\frac{1}{c} + \frac{1}{d} +\frac{1}{e}\right) \le 25 + 6\left(\sqrt{\frac {p}{q}} \minus{} \sqrt {\frac{q}{p}}\right)^2\] (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 |