## Task 4 - 220734

In the display window of a sports store, there is a stack of 550 balls of the same size. The stack consists of horizontal layers. Each layer contains balls arranged in a rectangular pattern, as shown in the image.

The numbers $a$ and $b$ are exactly 1 less in each layer compared to the corresponding numbers in the layer below.

In the bottom layer, 10 is the smaller of the two numbers $a, b$. In the top layer, 1 is the smaller of the two numbers $a, b$.

Determine the number of balls in the bottom layer from these details!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0464.jpg?height=286&width=620&top_left_y=2073&top_left_x=1138)
Let $b=10$ be the smaller of the two numbers $a, b$ and $a=x$ the larger in the bottom layer.

Then in the following layers, $b=9, b=8, \ldots, b=1$ are the smaller numbers and $a=x-1, a=x-2$, ..., $a=x-9$ are the larger of the two numbers $a, b$. This leads to

$$
\begin{gathered}
550=10 x+9(x-1)+8(x-2)+7(x-3)+6(x-4)+5(x-5)+4(x-6)+3(x-7)+2(x-8)+(x-9) \\
=10 x+9 x+8 x+7 x+6 x+5 x+4 x+3 x+2 x+x-9-16-21-24-25-24-21-16-9 \\
=55 x-165
\end{gathered}
$$

Solving for $x$, we get $55x - 165 = 550$, which simplifies to $55x = 715$ and $x = 13$.

Therefore, in the bottom layer, there are $10 \cdot 13$ balls, i.e., 130 balls.