Difference between revisions of "1964 IMO Problems/Problem 2"
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This is true by AM-GM. We can work backwards to get that the original inequality is true. | This is true by AM-GM. We can work backwards to get that the original inequality is true. | ||
+ | |||
+ | ==Solution 2== | ||
+ | Rearrange to get | ||
+ | <cmath>a(a-b)(a-c) + b(b-a)(b-c) + c(c-a)(c-b) \ge 0,</cmath> | ||
+ | which is true by Schur's inequality. | ||
+ | |||
+ | == See Also == {{IMO box|year=1964|num-b=1|num-a=3}} |
Revision as of 12:47, 29 January 2021
Contents
Problem
Suppose are the sides of a triangle. Prove that
Solution
We can use the substitution , , and to get
This is true by AM-GM. We can work backwards to get that the original inequality is true.
Solution 2
Rearrange to get which is true by Schur's inequality.
See Also
1964 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |