Question

If \[ x = \frac{x}{y+z}, \quad y = \frac{y}{z+x}, \quad z = \frac{z}{x+y} \] then which of the following statements is/are true?

• A. I and II
• B. I and III
• C. II and III
• D. None of these

Hint:

When given symmetric conditions, try substituting and verifying logical contradictions with simple values.

Answer 1

The Correct Option is D

Given: \[ x = \frac{x}{y+z} \Rightarrow y+z = 1
y = \frac{y}{z+x} \Rightarrow z+x = 1
z = \frac{z}{x+y} \Rightarrow x+y = 1 \] Add all three: \[ (y+z) + (z+x) + (x+y) = 3 \Rightarrow 2(x + y + z) = 3 \Rightarrow x + y + z = \frac{3}{2} \] Now check if all options hold — none of the algebraic identities are universally true. Test values show contradiction. Hence: \[ \boxed{\text{None of these}} \]