Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1991 IMO Problems/Problem 5"

(→‎Solution)
(→‎Solution)
Line 3: Line 3:
  
 
== Solution ==
 
== Solution ==
−
Let <math>A_{1}</math> , <math>A_{2}</math>, and <math>A_{3}</math> = <math>\measuredangle CAB</math>, <math>\measuredangle ABC</math>, <math>\measuredangle BCA</math>, respcetively.
+
Let <math>A_{1}</math> , <math>A_{2}</math>, and <math>A_{3}</math> be <math>\measuredangle CAB</math>, <math>\measuredangle ABC</math>, <math>\measuredangle BCA</math>, respcetively.
  
−
Let <math>\alpha_{1}</math> , <math>\alpha_{2}</math>, and <math>\alpha_{3}</math> = <math>\measuredangle PAB</math>, <math>\measuredangle PBC</math>, <math>\measuredangle PCA</math>, respcetively.
+
Let <math>\alpha_{1}</math> , <math>\alpha_{2}</math>, and <math>\alpha_{3}</math> be <math>\measuredangle PAB</math>, <math>\measuredangle PBC</math>, <math>\measuredangle PCA</math>, respcetively.
  
  
 
{{alternate solutions}}
 
{{alternate solutions}}

Revision as of 11:12, 12 November 2023

Problem

Let $\,ABC\,$ be a triangle and $\,P\,$ an interior point of $\,ABC\,$. Show that at least one of the angles $\,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $30^{\circ }$.

Solution

Let $A_{1}$ , $A_{2}$, and $A_{3}$ be $\measuredangle CAB$, $\measuredangle ABC$, $\measuredangle BCA$, respcetively.

Let $\alpha_{1}$ , $\alpha_{2}$, and $\alpha_{3}$ be $\measuredangle PAB$, $\measuredangle PBC$, $\measuredangle PCA$, respcetively.


Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1991_IMO_Problems/Problem_5&oldid=202569"