2002 AMC 12B Problem 19

Attempt Problem 19 of the 2002 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 12B solutions, or check the answer key.

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19.

a,

b,

and

c

are positive real numbers such that

a(b+c)=152,

b(c+a)=162,

and

c(a+b)=170,

then

abc

672

688

704

720

750

Add all three equations to get

ab+bc+ca

Subtract each original equation from that total to find

ab,

bc,

ca,

then multiply and take a square root

Adding the three equations gives

2(ab+bc+ca)=484,

ab+bc+ca=242.

Subtracting each original equation from this yields

bc=90,

ca=80,

and

ab=72.

Multiplying,

(abc)^2=90\cdot80\cdot72=720^2,

and since

abc\gt0,

we get

abc=720.

Thus, the correct answer is D.

Problem 19 in other years