Problem 6. Inside the magician's hat, there live 100 rabbits: white, blue, and green. It is known that if 81 rabbits are randomly pulled out of the hat, there will definitely be three of different colors among them. What is the minimum number of rabbits that need to be taken out of the hat to ensure that there are definitely two of different colors?

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# Answer: 61.

Solution. We will prove that if 61 rabbits are randomly pulled out of the hat, then among them there will be two of different colors. Suppose the opposite: let there be $a \geqslant 61$ rabbits of some color (for example, white). Let the second color by number of rabbits be blue. Then there are at least $\frac{100-a}{2}$ blue rabbits in the hat. Therefore, the total number of white and blue rabbits is at least

$$
a+\frac{100-a}{2}=\frac{100+a}{2} \geqslant \frac{161}{2}=80.5
$$

Since the number of rabbits is an integer, the total number of white and blue rabbits is at least 81, which contradicts the condition.

We will show that 60 rabbits may not be enough. Suppose there are 60 white rabbits and 20 blue and 20 green rabbits in the hat. Then it is possible that all the pulled-out rabbits are white. On the other hand, if 81 rabbits are pulled out, then among them there will definitely be rabbits of all three colors.

## Criteria

4 points. Correct solution.

3 points. Proven that 61 rabbits are enough.

1 point. Shown that 60 rabbits may not be enough.

0 points. Only the correct answer.