Algebra (Olympiad)hardFree

Algebra (Olympiad): Number Real Values Satisfying Equations

JEE Maths question with a full step-by-step solution.

Question

The number of real values of

x, y, z

for satisfying the equations

\sqrt{x^2 - y} = z - 1, \qquad \sqrt{y^2 - z} = x - 1, \qquad \sqrt{z^2 - x} = y - 1.

Solution

Answer: 1

Step 1: Each square root is non-negative, so

z \ge 1

x \ge 1

y \ge 1

. Squaring each equation,

x^2 - y = z^2 - 2z + 1, \qquad y^2 - z = x^2 - 2x + 1, \qquad z^2 - x = y^2 - 2y + 1.

Step 2: Add the three equations.

(x^2 + y^2 + z^2) - (x + y + z) = (x^2 + y^2 + z^2) - 2(x + y + z) + 3.

Step 3: Simplify.

-(x + y + z) = -2(x + y + z) + 3 \implies x + y + z = 3.

Step 4: Rewrite as

(x - 1) + (y - 1) + (z - 1) = 0

. Since each of

x - 1, y - 1, z - 1

is non-negative, the sum is zero only if each is zero, that is

x = y = z = 1

. Answer:

(x, y, z) = (1, 1, 1)

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