1964 AMC 12 Problem 32

Question

If

\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a},

then:

a

must equal

c

a+b+c+d

must equal zero

either

a=c

or

a+b+c+d=0,

or both

a+b+c+d\ne0

if

a=c

a(b+c+d)=c(a+b+d)

Cross-multiply the two fractions

Move all terms to one side and factor out

a-c

Cross-multiplication gives

(a+b)(a+d)=(b+c)(c+d).

Subtracting the right side and factoring yields

(a-c)(a+b+c+d)=0.

Hence either

a=c

or the sum is zero, and both may occur.

Therefore, the correct answer is C.

Problem 32 in other years