Evaluate the symmetric expression for nearby integers: If x = 997, y = 998, and z = 999, compute the value of x^2 + y^2 + z^2 − xy − yz − zx.

Introduction / Context:This problem tests recognition of a standard symmetric identity that simplifies expressions of the form x^2 + y^2 + z^2 − xy − yz − zx. Using the identity avoids large-number squaring and reduces the task to simple differences.

Given Data / Assumptions:

• x = 997
• y = 998
• z = 999
• Target expression: x^2 + y^2 + z^2 − xy − yz − zx

Concept / Approach:Use the identity: x^2 + y^2 + z^2 − xy − yz − zx = 1/2 * [ (x − y)^2 + (y − z)^2 + (z − x)^2 ]. This identity is derived by expanding the squares and collecting terms, and it dramatically simplifies computation when the numbers are close together.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 9, 16, 4: These do not match the identity-based calculation using true differences −1, −1, and 2.

Common Pitfalls:

• Forgetting the 1/2 factor in the identity.
• Sign mistakes in differences, though squares remove signs; the only risk is miscomputing 2^2.

Final Answer: