2021 AMC 10B Fall Problem 12

Question

Which of the following conditions is sufficient to guarantee that integers

x,

y,

and

z

satisfy the equation

x(x-y)+y(y-z)+z(z-x)

= 1?

x \gt y

and

y=z

x=y-1

and

y=z-1

x=z+1

and

y=x+1

x=z

and

y-1=x

x+y+z=1

Rewrite the expression using squared differences

The three squared differences must sum to

2

Let

E=x(x-y)+y(y-z)

+z(z-x).

Expanding gives

\begin{aligned} 2E={}&(x-y)^2+(y-z)^2\\ &+(z-x)^2. \end{aligned}

For the value to be

1,

the three nonnegative square terms must sum to

2.

Since

x,y,z

are integers, this means the squared differences are

1,1,0.

Thus two of the variables must be equal, and the third must differ from them by

1.

The condition

x=z

and

y-1=x

guarantees exactly that.

Thus, the answer is D .

Problem 12 in other years