USAAAO 2026, I кръг — Задача 26

• C

Solution: Since star X is larger, then during the eclipses either star Y is fully blocked (when star X is in front), or a portion of star X equal in area to star Y is blocked (when star Y is in front).

During the primary eclipse, the fraction of light blocked is $1 - 10^{-0.4\cdot0.200} = 0.168$, and during the secondary eclipse, the fraction of light blocked is $1 - 10^{-0.4\cdot0.100} = 0.0880$.

Since the areas eclipsed are the same, we know that the ratio of the surface brightness of the brighter star to the dimmer star is 0.168/0.0880, so the ratio of surface temperature of the brighter star to the dimmer star is $(0.168/0.0880)^{1/4}$.

Since we don’t know which of star X and star Y is brighter, $T_X/T_Y$ can be either $\sqrt[4]{\frac{0.168}{0.0880}} = 1.176$ or $\sqrt[4]{\frac{0.0880}{0.168}} = 0.850$, of which the first value is an answer choice.

Answer: C