To find the complex number $w$ given the conditions:

1. $z^2 + zw^* = 50$
2. $2z^* = w^*(1 - i)$
3. $\text{Re}(z) > 0$

we will follow a step-by-step approach to solve for $w$.

Step 1: Analyze the Equations

Given the complex numbers $z$ and $w$, we denote them as: $z = x + yi, \quad w = u + vi$ where $x, y, u,$ and $v$ are real numbers.

The conjugates $z^*$ and $w^*$ are: $z^* = x - yi, \quad w^* = u - vi$

Substitute these into the given equations.

Step 2: Substitute in the Second Equation

The second equation is: $2z^* = w^*(1 - i)$ Substitute the conjugates: $2(x - yi) = (u - vi)(1 - i)$

Expand and equate real and imaginary parts: $2x - 2yi = (u - vi)(1 - i) = (u - v - (u + v)i)$

Equating real and imaginary parts: $2x = u - v, \quad -2y = -u - v$

Rewriting these equations, we get: $u = x + y, \quad v = x - y$

Step 3: Substitute in the First Equation

Now use these in the first equation: $z^2 + zw^* = 50$

Compute $z^2$ and $zw^*$: $z^2 = (x + yi)^2 = x^2 - y^2 + 2xyi$ $zw^* = (x + yi)(u - vi) = (x + yi)((x + y) - (x - y)i)$

Expand $zw^*$: $zw^* = x(x + y) + y(x - y) + i(y(x + y) - x(x - y))$

Simplify $zw^*$: $zw^* = (x^2 + xy + yx - y^2) + i(yx + y^2 - x^2 + xy)$ $zw^* = (x^2 - y^2 + 2xy) + i(-x^2 + y^2 + 2xy)$

Thus, the equation $z^2 + zw^* = 50$ becomes: $(x^2 - y^2 + 2xy) + (x^2 - y^2 + 2xy) = 50$ $2(x^2 - y^2 + 2xy) = 50$

Simplify: $x^2 - y^2 + 2xy = 25$

Step 4: Solve for $z$ and $w$

Now use the real numbers: $x^2 + y^2 = r^2, \quad xy = s$ where $r^2 = x^2 + y^2$ and $s = xy$.

Thus, $x^2 - y^2 + 2xy = 25$, substitute: $r^2 + 2s = 25$ and for the previous solution $r = 5$, thus $x = y = \frac{5}{\sqrt{2}}$.

Now find $w$: $w = x + y + i(x - y) = 5 + i \times 0$

Thus, $w = 5$.

Solution:

The value of $w$ is: $\boxed{5}$