3. Let $\mathrm{P}_{1}, \mathrm{P}_{2}, \ldots, \mathrm{P}_{41}$ be 41 distinct points on the segment $\mathrm{BC}$ of a triangle $\mathrm{ABC}$, where $\mathrm{AB}=\mathrm{AC}$ $=7$. Evaluate the sum $\sum_{i=1}^{41}\left(\mathrm{AP}_{i}^{2}+\mathrm{P}_{i} \mathrm{~B} \cdot \mathrm{P}_{i} \mathrm{C}\right)$.
3. Answer: 2009

Construct a circle $\omega$ centred at $\mathrm{A}$ with radius $\mathrm{AB}=\mathrm{AC}=7$. The power of $\mathrm{P}_{i}$ with respect to $\omega$ is $\mathrm{P}_{i} \mathrm{~A}^{2}-7^{2}$, which is also equal to $-\mathrm{BP}_{f} \mathrm{P} C$. Thus, $\mathrm{P}_{f} \mathrm{~A}^{2}+\mathrm{BP}_{r} \cdot \mathrm{PC}=7^{2}=49$. Therefore, the sum equals $49 \times 41=2009$.