USAAAO 2022, I кръг — Задача 18

• C

If the smaller star is brighter, then the primary eclipse occurs when the smaller star is eclipsed, and the secondary eclipse occurs when the larger star is eclipsed. Let $A$ and $B$ denote the fractions of the brightness of the total system from the smaller and larger stars, respectively. Note that as defined $A + B = 1$. The increase in apparent magnitude during the primary eclipse is 1.5, so we have

Thus, $A = 0.75$. The increase in apparent magnitude during the secondary eclipse is 0.2. Let $k$ be the requested ratio of radii. Note that $(1 - k^2)$ of the larger star is visible during the secondary eclipse, so

This implies $1 - k^2 = 0.32$, and $k = 0.82$.

Answer: C