Difference between revisions of "1980 USAMO Problems/Problem 5"
(Created page with "== Problem == If <math>x, y, z</math> are reals such that <math>0\le x, y, z \le 1</math>, show that <math>\frac{x}{y + z + 1} + \frac{y}{z + x + 1} + \frac{z}{x + y + 1} \le 1 ...") |
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== Solution == | == Solution == | ||
− | {{ | + | Rewrite the given inequality so that <math>1</math> is isolated on the right side. Set the left side to be <math>f(x, y, z)</math>. Now a routine computation shows |
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+ | <math>\frac{\partial^2 f}{\partial x^2} = \frac{2y}{(x + z + 1)^3} + \frac{2z}{(x + y + 1)^3}\geq 0 </math> | ||
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+ | which shows that <math>f</math> is convex (concave up) in all three variables. Thus the maxima can only occur at the endpoints, i.e. if and only if <math>x, y, z \in \{0,1\}</math>. Checking all eight cases shows that the value of the expression cannot exceed 1. | ||
== See Also == | == See Also == | ||
{{USAMO box|year=1980|num-b=4|after=Last Question}} | {{USAMO box|year=1980|num-b=4|after=Last Question}} | ||
+ | {{MAA Notice}} | ||
[[Category:Olympiad Inequality Problems]] | [[Category:Olympiad Inequality Problems]] |
Latest revision as of 18:11, 3 July 2013
Problem
If are reals such that , show that
Solution
Rewrite the given inequality so that is isolated on the right side. Set the left side to be . Now a routine computation shows
which shows that is convex (concave up) in all three variables. Thus the maxima can only occur at the endpoints, i.e. if and only if . Checking all eight cases shows that the value of the expression cannot exceed 1.
See Also
1980 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.