B3. The perimeter of a rectangle is $4 \mathrm{~cm}$, and the angle between the diagonals is $60^{\circ}$. Draw a sketch of the rectangle with the angle between the diagonals marked. Calculate the lengths of both sides of this rectangle. The result should be exact and written in the form of simplified fractions with rationalized denominators. Solve the problem without using a pocket calculator.

## 16th Knowledge Competition

 in mathematics for students of secondary technical and vocational schoolsNational Competition, April 16, 2016

## Problems for 3rd Year

Time for solving: 120 minutes. In section A, we will award three points for each correct answer, and deduct one point for each incorrect answer. Enter your answers for section A in the left table, leave the right table blank.

![](https://cdn.mathpix.com/cropped/2024_06_07_7e68135e1a71f8c81168g-10.jpg?height=163&width=319&top_left_y=661&top_left_x=640)

| B1 | B2 | B3 |
| :--- | :--- | :--- |
|  |  |  |
B3.

![](https://cdn.mathpix.com/cropped/2024_06_07_7e68135e1a71f8c81168g-22.jpg?height=317&width=537&top_left_y=1812&top_left_x=702)
Sketch of the rectangle with the angle between the diagonals marked ..... 1 point
Considering $2 a+2 b=4$ ..... 1 point
Writing or considering $\tan 30^{\circ}=\frac{b}{a}$ ..... 1 point
Writing or considering $\tan 30^{\circ}=\frac{\sqrt{3}}{3}$ ..... 1 point
Solving the system of equations ..... $1^{*}$ point
Rationalizing the denominator ..... $1^{*}$ point
Calculating $a=(3-\sqrt{3}) \mathrm{cm}$ ..... 1 point
Calculating $b=(-1+\sqrt{3}) \mathrm{cm}$ ..... 1 point
Note: If the units are missing in the result, subtract 1 point.

## Third Year

![](https://cdn.mathpix.com/cropped/2024_06_07_7e68135e1a71f8c81168g-23.jpg?height=143&width=234&top_left_y=357&top_left_x=928)