Question

If three positive real numbers $x$, $y$ and $z$ satisfy $y - x = z - y$ and $x + y = 4$, then what is the minimum possible value of $y$?

• A. $2^{1/3}$
• B. $2^{2/3}$
• C. $3^{1/4}$
• D. $2^{4/3}$

Hint:

Use the method of substitution and symmetry to solve for the minimum or maximum values of variables in equations involving multiple variables.

Answer 1

The Correct Option is B

We are given that $y - x = z - y$, so we can express $z$ in terms of $x$ and $y$ as: \[ z = 2y - x \] We are also given that $x + y = 4$, so we can solve for $x$ as: \[ x = 4 - y \] Substituting this into the equation for $z$, we get: \[ z = 2y - (4 - y) = 3y - 4 \] Now, to minimize $y$, we use the condition that $x$, $y$, and $z$ must all be positive real numbers. By substituting the values of $x$, $y$, and $z$, we can minimize $y$.