2021 Fall AMC 12B Problem 7

Problem 7 of 25EasierAlgebra

Which of the following conditions is sufficient to guarantee that integers

x,

y,

and

z

satisfy the equation

\begin{aligned} &x(x - y) + y(y - z) \\ &\quad {}+ z(z - x) = 1? \end{aligned}

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Solution

The expression satisfies

\small \begin{aligned} &2\bigl[x(x-y) + y(y-z) + z(z-x)\bigr] \\ &= (x-y)^2 + (y-z)^2 + (z-x)^2. \end{aligned}

So the equation holds exactly when this sum of squares equals

2.

Since the three differences sum to

0,

this requires two of them to be

\pm 1

and one to be

0.

Option D gives

z - x = 0,

x - y = -1,

and

y - z = 1,

so the squares are

1 + 1 + 0 = 2.

This works for all such integers. Thus, the correct answer is D.