2015 Putnam A1

Let $A$ and $B$ be points on the same branch of the hyperbola $xy=1$. Suppose that $P$ is a point lying between $A$ and $B$ on this hyperbola, such that the area of the triangle $APB$ is as large as possible. Show that the region bounded by the hyperbola and the chord $AP$ has the same area as the region bounded by the hyperbola and the chord $PB$.

设$A$和$B$为双曲线$xy=1$同一分支上的点。假设$P$是该双曲线上$A$和$B$之间的点，使得三角形$APB$的面积尽可能大。证明由双曲线和弦 $AP$ 界定的区域与由双曲线和弦 $PB$ 界定的区域具有相同的面积。