Question

Consider integers \( x, y, z \). What is the minimum possible value of \( x^2 + y^2 + z^2 \)?
A: \( x + y + z = 89 \)
B: Among \( x, y, z \) two are equal.

• A. if the question can be answered using A alone but not B alone.
• B. if the question can be answered using B alone but not A alone.
• C. if the question can be answered using A and B together, but not using either A or B alone.
• D. if the question cannot be answered even using A and B together.
• E.

Hint:

When minimising sum of squares with fixed sum, equal distribution minimises value. Integer constraints may shift result slightly.

Answer 1

The Correct Option is C

A alone: infinitely many integer triples sum to 89; without constraints, can’t find min sum of squares. B alone: no info on actual values. Together: let \( x = y \), \( 2x + z = 89 \), minimise \( 2x^2 + z^2 \). By integer minimisation, solution found. Hence both together are needed.