1978 AMC 12 Problem 24

Attempt Problem 24 of the 1978 AMC 12 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 1978 AMC 12 solutions, or check the answer key.

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24.

If the distinct nonzero numbers

x(y-z),

y(z-x),

z(x-y)

form a geometric progression with common ratio

r,

then

r

satisfies the equation

r^2+r+1=0

r^2-r+1=0

r^4+r^2-1=0

(r+1)^4+r=0

(r-1)^4+r=0

Add the three given expressions

Write the three nonzero terms as

u,

ur,

ur^2

The three expressions have sum

\begin{aligned} &x(y-z)+y(z-x)\\ &\qquad+z(x-y)=0. \end{aligned}

Writing the nonzero geometric progression as

u,

ur,

ur^2

gives

u(1+r+r^2)=0.

Since

u\ne0,

r^2+r+1=0.

Therefore, the correct answer is A.

Problem 24 in other years