1969 Canadian MO Problems/Problem 10

Problem

Let $ABC$ be the right-angled isosceles triangle whose equal sides have length 1. $P$ is a point on the hypotenuse, and the feet of the perpendiculars from $P$ to the other sides are $Q$ and $R$ . Consider the areas of the triangles $APQ$ and $PBR$ , and the area of the rectangle $QCRP$ . Prove that regardless of how $P$ is chosen, the largest of these three areas is at least $2/9$ .

Solution

Let $AQ=x.$ Because triangles $APQ$ and $BPR$ both contain a right angle and a $45^\circ$ angle, they are isosceles right triangles. Hence, $PQ=RC=x$ and $QC=PR=BR=1-x.$

Now let's consider when $\frac13 <x<\frac23,$ or else one of triangles $APQ$ and $PBR$ will automatically have area greater than $\frac29.$ In this case, $[QCRP]>[ABC]-[APQ]-[PBR]>\frac29.$ Therefore, one of these three figures will always have area greater than $\frac29,$ regardless of where $P$ is chosen.

1969 Canadian MO (Problems)
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1969 Canadian MO Problems/Problem 10

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