Question

Let the radius of each circular park be $r$, and the distances to be traversed by the sprinters A, B, and C be $a$, $b$, and $c$ respectively. Which of the following is true?

• A. $b - a = c - b = \sqrt{3} r$
• B. $b - a = c - b = \sqrt{3} r$
• C. $b = \frac{a + c}{2} = (2 + \sqrt{3}) r$
• D. $c = 2b - a = (2 + \sqrt{3}) r$
• A. \( B_1 \), \( C_1 \)
• B. \( B_3 \), \( C_3 \)
• C. \( B_1 \), \( C_3 \)
• D. \( B_1 \), Somewhere between \( C_3 \) and \( C_1 \)
• A. \( A_2 \), \( C_3 \)
• B. \( A_3 \), \( C_3 \)
• C. \( A_3 \), \( C_2 \)
• D. Somewhere between \( A_2 \) and \( A_3 \), Somewhere between \( C_3 \) and \( C_1 \)

Answer 1

The Correct Option is D

We are given three circular parks with equal size, centered at $A_1$, $B_1$, and $C_1$ respectively. The distances traversed by the sprinters A, B, and C are labeled as $a$, $b$, and $c$ respectively. The three paths are formed by the intersections of the circles, creating triangular paths for each sprinter. From symmetry and geometry of the circles, we can deduce the following: The relationship between the distances traversed by the sprinters can be derived by considering the distances from each sprinter's starting point along their respective paths. Using geometric reasoning, the distances are related by: \[ c = 2b - a = (2 + \sqrt{3}) r \] Thus, the Correct Answer is option 4.

Answer 2

First, calculate the total time taken by A to finish her sprint. The total distance A covers is the sum of the three segments \( A_1 A_2 \), \( A_2 A_3 \), and \( A_3 A_4 \), but since the distances are not provided, we focus on the time taken by A, which is given by the formula: \[ t_A = \frac{d_1}{v_1} + \frac{d_2}{v_2} + \frac{d_3}{v_3} \] where \( v_1 = 20 \), \( v_2 = 30 \), and \( v_3 = 15 \). For B, her total time to cover the entire distance is: \[ t_B = \frac{d_B}{v_B} = \frac{d_B}{10\sqrt{3} + 20} \] Since B's speed is constant, we compare the ratio of \( t_A \) to \( t_B \) to determine where B is when A finishes her sprint. For C, the total time to cover her path is the sum of the times for each segment: \[ t_C = \frac{d_{C_1 C_2}}{v_{C_1 C_2}} + \frac{d_{C_2 C_3}}{v_{C_2 C_3}} + \frac{d_{C_3 C_4}}{v_{C_3 C_4}} \] where \( v_{C_1 C_2} = v_{C_2 C_3} = \frac{40}{3}(\sqrt{3} + 1) \) and \( v_{C_3 C_4} = 120 \). By comparing the times \( t_A \), \( t_B \), and \( t_C \), we can determine that when A finishes, B is at position \( B_1 \) and C is at position \( C_3 \). Thus, the Correct Answer is Option (3). \[ \boxed{B_1, C_3} \]

Answer 3

From the given information, the ratio \( u^2:v^2:w^2 \) corresponds to the ratio of the areas of triangles \( A_1 A_2 A_3 \), \( B_1 B_2 B_3 \), and \( C_1 C_2 C_3 \). This suggests that the distances traversed by the sprinters are proportional to the square roots of the areas. Let the areas of the triangles be denoted as \( \text{Area A} \), \( \text{Area B} \), and \( \text{Area C} \), which are proportional to \( u^2 \), \( v^2 \), and \( w^2 \), respectively. When sprinter B reaches \( B_3 \), we calculate where A and C are by using the proportionality of their speeds and the areas of the respective triangles: - The distance A has covered when B reaches \( B_3 \) is proportional to the ratio of \( u \) to \( v \), so A will be at \( A_2 \). - Similarly, C, being proportional to the ratio \( w \) to \( v \), will be at \( C_3 \). Thus, when B reaches \( B_3 \), sprinter A will be at \( A_2 \) and sprinter C will be at \( C_3 \). Therefore, the Correct Answer is Option (1). \[ \boxed{A_2, C_3} \]