Problem 5.6. In the zoo, there are red, yellow, and green parrots (there is at least one parrot of each of the listed colors; there are no parrots of other colors in the zoo). It is known that among any 10 parrots, there is definitely a red one, and among any 12 parrots, there is definitely a yellow one. What is the maximum number of parrots that can be in the zoo?
Answer: 19.

Solution. Let there be $x$ red, $y$ yellow, and $z$ green parrots in the zoo.

Since among any 10 parrots there is a red one, the number of non-red parrots does not exceed 9, that is, $y+z \leqslant 9$. By similar reasoning, we get that the number of non-yellow parrots does not exceed 11, that is, $x+z \leqslant 11$.

Thus,

$$
\begin{gathered}
x+y+2 z=(y+z)+(x+z) \leqslant 9+11=20 ; \\
x+y+z \leqslant 20-z \leqslant 19
\end{gathered}
$$

This means that the total number of parrots does not exceed 19. At the same time, there could have been 10 red, 8 yellow, and 1 green parrot in the zoo, that is, exactly 19 parrots.