Difference between revisions of "2001 USAMO Problems/Problem 3"
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− | Without loosing generality, we assume <math>(b-1)(c-1)\ge 0</math>. | + | Without loosing generality, we assume <math>(b-1)(c-1)\ge 0</math>. From the given equation, we can express <math>a</math> in the form <math>b</math> and <math>c</math>, |
+ | <center> <math>a=\frac{\sqrt{(4-b^2)(4-c^2)}-bc}{2} </math></center> | ||
== See also == | == See also == |
Revision as of 22:43, 8 February 2011
Problem
Let and satisfy
![$a^2 + b^2 + c^2 + abc = 4.$](http://latex.artofproblemsolving.com/1/6/e/16e70ab813b2e9287a1015d7b890d16f94a7073e.png)
Show that
![$ab + bc + ca - abc \leq 2.$](http://latex.artofproblemsolving.com/6/6/c/66c37e5bd5601bffeb016667562ca756e7dd1d9b.png)
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
Without loosing generality, we assume . From the given equation, we can express
in the form
and
,
![$a=\frac{\sqrt{(4-b^2)(4-c^2)}-bc}{2}$](http://latex.artofproblemsolving.com/4/3/4/4344f8b39fdd47d62fae63057b6575aef0c9e7a6.png)
See also
2001 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |