Just The Third Side

Relevant wiki: Area of Triangles - Problem Solving - Medium

The minimum perimeter occurs with $BC=3$ , and the least area of $BPC$ in this case occurs when $BP=4$ and $CP=2$ (the case $BP=3$ and $CP=1$ is not permitted). This gives $x =\tfrac34\sqrt{15}$ .

The maximum perimeter occurs when $BC=14$ . While a triangle $ABC$ is possible with $BC=15$ , no interior point $P$ can exist in that case. The maximum area of $BPC$ in this case occurs when $BP=8$ and $CP=7$ , giving $y = \tfrac14\sqrt{5655}$ . The very close case of $BP=9$ and $CP=6$ gives the slightly smaller area $\tfrac14\sqrt{5423}$ . The alternative vase of $BP=10$ and $CP=5$ gives an even smaller area. The case $BP=7$ and $CP=8$ is possible, giving the same maximum area $y$ .

Thus $x+y \; = \; \tfrac14\big[\sqrt{5655} + 3\sqrt{15}\big]$ making the answer $1+4+5655 + 3 + 15 = \boxed{5678}$ . Incidentally, $15$ is a factor of $5655$ , so a simpler form for $x+y$ is $\tfrac14\sqrt{15}\big[3 + \sqrt{377}\big]$