Difference between revisions of "1981 IMO Problems/Problem 1"
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We note that <math>BC \cdot PD + CA \cdot PE + AB \cdot PF</math> is twice the triangle's area, i.e., constant. By the [[Cauchy-Schwarz Inequality]], | We note that <math>BC \cdot PD + CA \cdot PE + AB \cdot PF</math> is twice the triangle's area, i.e., constant. By the [[Cauchy-Schwarz Inequality]], | ||
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| − | < | + | <cmath> |
{(BC \cdot PD + CA \cdot PE + AB \cdot PF) \left(\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF} \right) \ge ( BC + CA + AB )^2} | {(BC \cdot PD + CA \cdot PE + AB \cdot PF) \left(\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF} \right) \ge ( BC + CA + AB )^2} | ||
| − | </ | + | </cmath>, |
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| − | with equality exactly when <math>PD = PE = PF </math>, which occurs when <math>P </math> is the triangle's incenter, | + | with equality exactly when <math>PD = PE = PF </math>, which occurs when <math>P </math> is the triangle's incenter or one of the three excenters. But since we know <math>P </math> is inside <math>\triangle ABC </math>, we can say <math>P </math> is the incenter. <math>\square </math> |
{{alternate solutions}} | {{alternate solutions}} | ||
| − | {{IMO box|before=First question|num-a=2|year= | + | {{IMO box|before=First question|num-a=2|year=1981}} |
[[Category:Olympiad Geometry Problems]] | [[Category:Olympiad Geometry Problems]] | ||
Latest revision as of 12:06, 26 August 2024
Problem
is a point inside a given triangle 
. 
are the feet of the perpendiculars from to the lines 
, respectively. Find all for which
is least.
Solution
We note that 
is twice the triangle's area, i.e., constant. By the Cauchy-Schwarz Inequality,
![\[{(BC \cdot PD + CA \cdot PE + AB \cdot PF) \left(\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF} \right) \ge ( BC + CA + AB )^2}\]](http://latex.artofproblemsolving.com/7/4/a/74af4eb3cfa9b3a1c741e98e758a03ce0b61a235.png)
,
with equality exactly when 
, which occurs when is the triangle's incenter or one of the three excenters. But since we know is inside 
, we can say is the incenter. 
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
| 1981 IMO (Problems) • Resources | ||
| Preceded by First question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
| All IMO Problems and Solutions | ||
