WB PRE-RMO 2015 22nd November

Problem 1

Find the sum 𝑆=Σ2015𝑘=1(−1)𝑘(𝑘+1)2⋅𝑘

Problem 2

Suppose in $\triangle A B C$, $A B=\sqrt{3}$, $B C=1$, $C A=2$. Suppose there exists a point $P_{0}$ in the plane of $\triangle A B C$ such that $A P_{0}$+$B P_{0}$+$C P_{0} \leq A P+B P+C P$ for all points $P$ in the plane of $\triangle A B C$. Find $(\left.A P_{0}+B P_{0}+C P_{0}\right)^{2}$