B4. Calculate the height and angle $\alpha$ of an isosceles trapezoid given $a=15 \mathrm{~cm}, c=7 \mathrm{~cm}, b=d=$ $4 \sqrt{2} \mathrm{~cm}$.

## PROBLEMS FOR THE THIRD YEAR

Before you are two sets of problems. Problems 1 to 6 of the first set are solved by selecting the correct answer from the five proposed answers on this sheet and writing it in the table under the corresponding number. Only one answer is correct. A correct answer will be scored with two points, while a wrong answer will result in the deduction of one point.

Problems 1 to 4 of the second set are solved on the provided paper. The solution to each of these problems will be evaluated on a scale from 0 to 6 points. On the sheets where you will solve the problems, do not sign, write only your code. Write your solution clearly and legibly with a pen.

The time for solving is 90 minutes.

THE NATIONAL COMPETITION COMMITTEE WISHES YOU GREAT SUCCESS.

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-06.jpg?height=174&width=485&top_left_y=958&top_left_x=800)

## PART I
B4. We draw a sketch from which we read $\frac{a-c}{2}=4 \mathrm{~cm}$. We calculate the height: $v=\sqrt{(4 \sqrt{2})^{2}-4^{2}}=\sqrt{16}=4 \mathrm{~cm}$. Since $v=\frac{a-c}{2}$, the right triangle with the lengths of the legs $v$ and $\frac{a-c}{2}$ is isosceles. The angle $\alpha$ is therefore $45^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-13.jpg?height=315&width=760&top_left_y=859&top_left_x=1156)

Correct sketch 1 point

Determination of $\frac{a-c}{2}=4 \mathrm{~cm}$. 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-13.jpg?height=88&width=1642&top_left_y=1301&top_left_x=273)

Calculation of the angle $\alpha=45^{\circ}$ 1 point

Use of trigonometric function or Pythagorean theorem to calculate the height 1 point

Calculated height $v=4 \mathrm{~cm}$

1 point

## Third Year

I. PART

| Task | 1 | 2 | 3 | 4 | 5 | 6 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| Answer | $\mathrm{D}$ | $\mathrm{B}$ | $\mathrm{C}$ | $\mathrm{C}$ | $\mathrm{A}$ | $\mathrm{D}$ |