USAAAO 2024, I кръг — Задача 9

• D

Solution:

According to the figure below,

for a given small $\theta$ we can have: $\tan(\theta) \approx \theta \approx \frac{r}{d} \rightarrow \theta = \text{const.} = \frac{r_e}{d_e} = \frac{r_m}{d_m}$

$\frac{d_m}{d_e} = \frac{r_m}{r_e}$

Let N be the number of observed stars:

$N \propto d^3 \rightarrow \frac{N_m}{N_e} = (\frac{d_m}{d_e})^3 = (\frac{1.5}{1})^3 \approx 4.$

Answer: D