isi-entrance 2009 Q3

isi-entrance · India · solved Areas Between Curves Maximize or Optimize Area
A triangle has vertices $A$, $B$, $C$. A point $P$ is chosen on side $AB$, and lines through $P$ parallel to the other sides create smaller triangles $APQ$ and $BPR$ and a parallelogram $PQCR$. Find the minimum value of the maximum of the areas of triangles $APQ$ and $BPR$ as a fraction of the area of $ABC$.
Assume that by symmetry $P$ is closer to $A$ than to $B$. Then area of $APQ$ is less than $BPR$. So we disregard $APQ$.
If $P$ moves away from $A$, the area of $PQCR$ increases while the area $BPR$ decreases; therefore the minimum of their maximum occurs when the two areas are equal in size, which is when $$CR/RB = 1/2$$ so $CR/1 = 1/3$. Thus $M = 2/9$. 
A triangle has vertices $A$, $B$, $C$. A point $P$ is chosen on side $AB$, and lines through $P$ parallel to the other sides create smaller triangles $APQ$ and $BPR$ and a parallelogram $PQCR$. Find the minimum value of the maximum of the areas of triangles $APQ$ and $BPR$ as a fraction of the area of $ABC$.
Paper Questions
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