B4 Gardeners will set up a flower bed in the center of the city. The flower bed is square-shaped with a side length of 4 meters. They have decided to play with the planting of tulips. On the shaded part, they will plant bulbs of red tulips, and on the unshaded part, they will plant bulbs of white tulips (see the image).

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf2b29f7493f2ef25deg-4.jpg?height=417&width=417&top_left_y=1810&top_left_x=862)

A Calculate the area of the shaded and the unshaded part of the flower bed in $\mathrm{cm}^{2}$ accurately.

B In the nursery, tulip bulbs are packed in larger and smaller bags. A larger bag costs 13 EUR and contains 10 bulbs of red tulips and 20 bulbs of white tulips. The price of a smaller bag is 3 EUR, and it contains 5 bulbs of red tulips and 3 bulbs of white tulips. What is the price of a single bulb of a red tulip and a single bulb of a white tulip?

C How long would the new enlarged square flower bed be if the gardeners continued the drawn pattern for one more step?

## 11th Mathematics Knowledge Competition for Vocational School Students  National Competition, April 16, 2011

## Solutions to the Problems and Scoring

A contestant who arrives at the solution by any correct method (even if the scoring guide does not anticipate it) receives all possible points.

For a correct method, we consider any procedure that

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

$V$ in the table are the correct answers to the multiple-choice questions. Each correct answer is scored 2 points, an incorrect answer $-0.5$ points, and if the question is not answered, 0 points. To avoid a possible negative final score, each contestant is given 3 initial points.

| A1 | A2 | A3 | A4 | A5 | A6 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| A | E | D | D | C | B |
B4 The shaded part of the trellis can be composed of two circles with a radius of $0.5 \mathrm{~m}$ and two circles with a radius of $1 \mathrm{~m}$. The area of the shaded part, which has been planted with bulbs of red tulips, is equal to the sum of the areas of the two larger and two smaller circles: $S=2 \pi(0.5 \mathrm{~m})^{2}+2 \pi(1 \mathrm{~m})^{2}=7.85 \mathrm{~m}^{2}$. The entire trellis is square-shaped with an area of $(4 \mathrm{~m})^{2}=16 \mathrm{~m}^{2}$. The area of the unshaded part of the trellis, planted with bulbs of white tulips, is $16 \mathrm{~m}^{2}-7.85 \mathrm{~m}^{2}=8.15 \mathrm{~m}^{2}$.

Let $\mathrm{Z} x$ denote the price of a bulb for a red tulip, and $\mathrm{z} y$ for a white tulip. According to the text, we set up the system of equations:

$$
\begin{gathered}
10 x+20 y=13 \\
5 x+3 y=3
\end{gathered}
$$

From this, we calculate that $x=0.3$ EUR and $y=0.5$ EUR.

If the gardeners continued the pattern for 1 more step, the trellis would be $8 \mathrm{~m}$ long.

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf2b29f7493f2ef25deg-7.jpg?height=68&width=1662&top_left_y=1325&top_left_x=254)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf2b29f7493f2ef25deg-7.jpg?height=57&width=1602&top_left_y=1385&top_left_x=313)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf2b29f7493f2ef25deg-7.jpg?height=57&width=1648&top_left_y=1451&top_left_x=267)

The price of a bulb of a white tulip is 0.5 EUR. ................................................. 1 t

C The length of the new trellis would be $8 \mathrm{~m}$ ............................................................ 1 t