Question

Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a,$ $b,$ $c,$ and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a+b+c=43$ and $d=3.$

Answer

Answer: 441

Concepts:mass points ratio and proportion symmetry (algebra)

Difficulty rating: 2380

Small Hint:

Express the three barycentric coordinates of $P$ using the ratios $a:d,$ $b:d,$ $c:d$

Big Hint:

Use $\frac3{a+3}+\frac3{b+3}+\frac3{c+3}=1$ and expand symmetrically

Solution:

Along the cevian from $A,$ the barycentric coordinate at $A$ is $\frac{d}{a+d},$ and similarly at the other vertices. Since the three coordinates sum to $1$ and $d=3,$ $\frac3{a+3}+\frac3{b+3}+\frac3{c+3}=1.$ Put $s_1=a+b+c=43,$ $s_2=ab+bc+ca,$ and $s_3=abc.$ Clearing denominators gives $\begin{aligned}3(s_2+6s_1+27)&=s_3+3s_2\\&\quad+9s_1+27.\end{aligned}$ Therefore $s_3=9s_1+54,$ so $s_3=9(43)+54=441.$

1988 AIME Problem 12

Answer: 441

Difficulty rating: 2380

Small Hint:

Big Hint:

Solution:

Problem 12 in Other Years