B3. Let $A B C D E F$ be a regular hexagon, $P$ the midpoint of side $A B$ and $R$ the midpoint of side $E F$, as shown in the figure. What is the ratio of the area of quadrilateral $A P R F$ to the area of quadrilateral $B C D P$? (6 points)

![](https://cdn.mathpix.com/cropped/2024_06_07_6340f9febbabfa1859cdg-13.jpg?height=334&width=377&top_left_y=181&top_left_x=1542)

## Problems for 4th Grade

Time for solving: 120 minutes. In section A, a correct answer will be awarded two points, while a wrong answer will result in a deduction of one point. Enter your answers for section A in the left table.
![](https://cdn.mathpix.com/cropped/2024_06_07_6340f9febbabfa1859cdg-14.jpg?height=166&width=802&top_left_y=534&top_left_x=640)
B3. Let's denote the side length of a regular hexagon by $a$. The quadrilateral $A P R F$ is a trapezoid with a height of $\frac{1}{2} \cdot \frac{a \sqrt{3}}{2}=\frac{a \sqrt{3}}{4}$, so its area is $P_{A P R F}=\frac{a \sqrt{3}}{4} \cdot \frac{|P R|+|A F|}{2}=$ $\frac{a \sqrt{3}\left(\frac{3}{2} a+a\right)}{8}=\frac{5 \sqrt{3} a^{2}}{16}$. The triangle $P B D$ has a side length of $|B D|=2 \cdot \frac{a \sqrt{3}}{2}=\sqrt{3} a$ and a height of $|P B|=\frac{a}{2}$, so its area is $P_{P B D}=\frac{\frac{a}{2} \cdot \sqrt{3} a}{2}=\frac{\sqrt{3} a^{2}}{4}$. The triangle $B C D$ has a side length of $|B D|=\sqrt{3} a$ and a height of $\frac{a}{2}$, so its area is also $P_{B C D}=$ $\frac{\sqrt{3} a^{2}}{4}$. The area of the quadrilateral $B C D P$ is therefore $P_{B C D P}=P_{P B D}+P_{B C D}=2 \cdot \frac{\sqrt{3} a^{2}}{4}=\frac{\sqrt{3} a^{2}}{2}$. The ratio of the areas is $\frac{P_{A P R F}}{P_{B C D P}}=\frac{5}{8}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_6340f9febbabfa1859cdg-24.jpg?height=69&width=1579&top_left_y=1553&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_6340f9febbabfa1859cdg-24.jpg?height=60&width=1576&top_left_y=1606&top_left_x=274)

![](https://cdn.mathpix.com/cropped/2024_06_07_6340f9febbabfa1859cdg-24.jpg?height=60&width=1579&top_left_y=1655&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_6340f9febbabfa1859cdg-24.jpg?height=58&width=1579&top_left_y=1710&top_left_x=273)

The calculated ratio of the areas .......................................................................

## Solutions for 4th grade

In section A, a correct answer will be awarded two points, while an incorrect answer will result in the deduction of one point. To avoid a potential negative final score, each contestant is given an initial score of 3 points.

| 1 | 2 | 3 |
| :--- | :--- | :--- |
| $\mathrm{D}$ | $\mathrm{B}$ | $\mathrm{D}$ |

Justifications: