3. Mario brought a round pizza with an area of $4 \mu^{2}$ and cut it into 4 pieces with two perpendicular straight cuts. Each cut passed 50 cm from the center of the pizza. His brother Luigi took the largest and the smallest piece, while the other two pieces went to Mario. Find the total area of the pieces that Mario received.
Answer: $1.5 m^{2}$.

## Solution:

Let's make two more cuts, symmetrically (relative to the center) to the ones already made. We will denote the areas of some regions as shown in the figure on the right (the shaded area is what Mario received). Then we notice that Mario received pieces with areas $x+y$ and $x+y$ (a total of $S=2x+2y$), while Luigi received a small piece with area $x$ and a large piece consisting of two parts $y$, one part $x$, and the central square with a side of $1 m$ (2 times $50 \text{ cm}$) - that is, a total area of $2x+2y+1 m^{2}=S+1 m^{2}$. Together, they ate the whole pizza, so $S+(S+1 m^{2})=4 m^{2}$, from which $S=1.5 m^{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_71062315d0492b8e06a2g-3.jpg?height=434&width=420&top_left_y=648&top_left_x=1503)

## Criteria:

Correct answer without correct justification - 1 point.

It is shown that the pieces that Luigi received can be composed of the pieces that Mario received, plus the central square - 3 points.

The correct idea of the solution, but an incorrect answer due to an incorrect calculation of the area of the central square or an arithmetic error in solving the equation - 4 points.