1991 IMO Problems/Problem 5

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 $\A_{1}$ (Error compiling LaTeX. Unknown error_msg) , $\A_{2}$ (Error compiling LaTeX. Unknown error_msg), and $\A_{3}$ (Error compiling LaTeX. Unknown error_msg) = $\measuredangle CAB$ , $\measuredangle ABC$ , $\measuredangle BCA$ , respcetively.

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

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

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1991 IMO Problems/Problem 5

Problem

Solution