(a) Let's first find out how much magical syrup is in 300 mL of blue potion. Since the blue potion is 15% magical syrup by volume, the amount of magical syrup in 300 mL of blue potion is 0.15 x 300 = 45 mL.

Now, let's assume that we add x mL of red potion to this mixture. The resulting potion will have a total volume of 300 + x mL, and its magical syrup content will be (45 mL + 0.5x mL) / (300 mL + x mL). We want this to be equal to 20% (or 0.2), so we can set up the following equation:

(45 mL + 0.5x mL) / (300 mL + x mL) = 0.2

Solving for x, we get:

x = 150 mL

Therefore, 150 mL of red potion must be added to 300 mL of blue potion to produce a potion that is 20% magical syrup by volume.

(b) Let's assume that we mix x mL of red potion with y mL of blue potion to produce 180 mL of a potion that is 40% magical syrup by volume. We can set up the following two equations based on the amount of magical syrup and the total volume:

Amount of magical syrup: 0.5x mL + 0.15y mL = 0.4(180 mL)

Total volume: x mL + y mL = 180 mL

Simplifying the first equation, we get:

0.5x + 0.15y = 72

We can now use substitution to solve for x and y. Solving the second equation for y, we get:

y = 180 - x

Substituting this into the first equation, we get:

0.5x + 0.15(180 - x) = 72

Solving for x, we get:

x = 120 mL

Substituting this into the equation y = 180 - x, we get:

y = 60 mL

Therefore, 120 mL of red potion and 60 mL of blue potion can be combined to produce 180 mL of a potion that is 40% magical syrup by volume.

(c) Let's assume that we mix x mL of red potion with y mL of blue potion to produce a potion that is 25% magical syrup by volume. We can set up the following two equations based on the amount of magical syrup and the total volume:

Amount of magical syrup: 0.5x mL + 0.15y mL = 0.25(x+y) mL

Total volume: x mL + y mL = some value

We have two equations and two unknowns, so we can solve for x and y. Simplifying the first equation, we get:

0.25x - 0.1y = 0

Substituting the second equation into this equation, we get:

0.25x - 0.1(x + y) = 0

Simplifying, we get:

0.15x - 0.1y = 0

We can now use substitution to solve for x and y. Solving the second equation for y, we get:

y = some value - x

Substituting this into the first equation, we get:

0.15x - 0.1(some value - x) = 0

Simplifying, we get:

0.25x - 0.1(some value) = 0

Therefore, we can see that there is no solution that satisfies these equations for any positive values of x and y. So there is no combination of red