Difference between revisions of "2001 USAMO Problems/Problem 3"
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== Solution == | == Solution == | ||
− | + | First we prove the lower bound. | |
+ | |||
+ | Note that we cannot have <math>a, b, c</math> all greater than 1. | ||
+ | Therefore, suppose <math>a \le 1</math>. | ||
+ | Then | ||
+ | <cmath>ab + bc + ca - abc = a(b + c) + bc(1-a) \ge 0.</cmath> | ||
Without loss of generality, we assume <math>(b-1)(c-1)\ge 0</math>. From the given equation, we can express <math>a</math> in terms of <math>b</math> and <math>c</math>, | Without loss of generality, we assume <math>(b-1)(c-1)\ge 0</math>. From the given equation, we can express <math>a</math> in terms of <math>b</math> and <math>c</math>, | ||
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This completes the proof. | This completes the proof. | ||
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== See also == | == See also == |
Revision as of 17:49, 31 May 2011
Problem
Let and satisfy

Show that

Solution
First we prove the lower bound.
Note that we cannot have all greater than 1.
Therefore, suppose
.
Then
Without loss of generality, we assume . From the given equation, we can express
in terms of
and
,

Thus,

From Cauchy,

This completes the proof.
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 |