(given to Amélie Héliou and Adrien Laroche). On a deserted island, there are 34 chameleons. Initially, 7 are yellow, 10 are red, and 17 are green. When two chameleons of different colors meet, they both change to the third color. When two chameleons of the same color meet, nothing happens. After a year, all the chameleons have become the same color. Which one?
. Green. Indeed, the difference between the number of yellow chameleons and the number of red chameleons is always a multiple of 3: if a yellow chameleon meets a green one, this difference decreases by 3, as there is one less yellow and two more reds; if a red meets a green, it increases by 3. However, if a yellow meets a red, the difference remains unchanged, as the number of yellows and the number of reds each decrease by one. And if two chameleons of the same color meet, the difference also remains unchanged. Since, at the start, the difference 7 - 10 = -3 is a multiple of 3, it will always be a multiple of 3: the same cannot be said of the difference between the number of greens and the number of reds, as this, at the start, is not a multiple of 3: 17 - 10 = 7 is of the form 3k + 1, so it will always remain of this form. The same applies to the difference between the number of greens and the number of yellows. If there were 34 yellows and 0 reds, the difference between the number of yellows and the number of reds would not be a multiple of 3; the same applies if there were 34 reds and 0 yellows. Therefore, the color of all the chameleons after a year can only be green. To prove that it is possible for all the chameleons to turn green, it is necessary to find a sequence of encounters that leads to this situation: if the 7 yellows meet 7 of the reds, there will be 3 reds and 31 greens left. One of the greens meets a red, and there will be 2 yellows, 2 reds, and 30 greens left. Then the two reds meet the two yellows.