B3. The side of the first square measures $8 \mathrm{~cm}$, the side of the second square measures $12 \mathrm{~cm}$, and the perimeter of the third square is one quarter less than the sum of the perimeters of the first two squares. The squares are placed next to each other as shown in the figure.

(a) How many cm is the perimeter of the first and how many the perimeter of the second square?

![](https://cdn.mathpix.com/cropped/2024_06_07_b93fa657b92bc897cb7bg-09.jpg?height=320&width=605&top_left_y=183&top_left_x=1294)

(b) How many cm is the length of the side of the third square?

(c) By how many centimeters is point $C$ higher than point $B$ relative to the horizontal?

(d) How many ${ }^{\circ}$ does angle $\alpha$ measure?

(e) The largest square is cut into smaller, congruent squares with integer side lengths so that there is no waste. How many smaller squares do we get and what is the length of their side? Present all solutions.

## 21st Mathematics Knowledge Competition for Vocational School Students  National Competition, May 15, 2021

## Solutions to the tasks for 1st and 2nd year students

| $\mathrm{A} 1$ | $\mathrm{~A} 2$ | $\mathrm{~A} 3$ | $\mathrm{~A} 4$ | $\mathrm{~A} 5$ | $\mathrm{~A} 6$ | $\mathrm{~A} 7$ | $\mathrm{~A} 8$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| A | B | B | C | A | C | C | D |
B3. The perimeter of the first square measures $o_{1}=4 \cdot 8=32 \mathrm{~cm}$, and the perimeter of the second square measures $o_{2}=4 \cdot 12=$ $48 \mathrm{~cm}$.

The perimeter of the third square measures $o_{3}=\frac{3}{4} \cdot(32+48)=60 \mathrm{~cm}$. The side length of the third square measures $a_{3}=\frac{60}{4}=15 \mathrm{~cm}$.

Point $C$ is $15-8=7 \mathrm{~cm}$ higher than point $B$.

The angle $\alpha$ is calculated using the trigonometric function $\tan \alpha=\frac{15}{35}=0.429$. From this, we get $\alpha=23^{\circ} 11^{\prime}$.

There are three possible solutions. We get: (i) 225 squares with a side length of $1 \mathrm{~cm}$, (ii) 25 squares with a side length of $3 \mathrm{~cm}$, (iii) 9 squares with a side length of $5 \mathrm{~cm}$.

(a) Calculation of the perimeter of the first square: $o_{1}=32 \mathrm{~cm}$ and the second square $o_{2}=48 \mathrm{~cm}$. ........ $1 \mathrm{t}$

(b) Calculation of the side length of the third square $a_{3}=15 \mathrm{~cm} . \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots \mathrm{t}$

(c) Calculation of the distance by which point $C$ is higher than point $B: 7 \mathrm{~cm} . \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots$ t

(d) Use of the correct trigonometric function tangent and calculation of the angle $\alpha=25.78^{\circ} \ldots \ldots \ldots \ldots \ldots \ldots . .2$ t

![](https://cdn.mathpix.com/cropped/2024_06_07_b93fa657b92bc897cb7bg-14.jpg?height=60&width=1739&top_left_y=2277&top_left_x=176)

Presentation of the remaining two solutions. ............................................................. $1 \mathrm{t}$