12. Target (from 8th grade, 2 points). A target is hanging on the wall, consisting of five zones: a central circle (the bullseye) and four colored rings (see figure). The width of each ring is equal to the radius of the bullseye. It is known that the number of points for hitting each zone is inversely proportional to the probability of hitting that zone, and that the bullseye is worth 315 points. How many points is a hit in the blue (second-to-last) zone worth?
Solution. Suppose, for definiteness, that the apple has a radius of 1. Then the blue zone is enclosed between circles with radii 3 and 4. The probability of hitting the blue zone relative to the probability of hitting the apple is the ratio of the areas of these zones:

$$
\frac{p_{g}}{p_{c}}=\frac{1}{4^{2}-3^{2}}=\frac{1}{7}
$$

Then the number of points \( x \) for the blue zone can be found from the proportion

$$
\frac{x}{315}=\frac{p_{g}}{p_{c}}=\frac{1}{7}
$$

from which \( x = 315 : 7 = 45 \).

Answer: 45.

## $\mathrm{XI}$ Internet Olympiad in Probability Theory and Statistics. Main (correspondence) round. $15.12 .77-21.01 .18$