Maths Olympiad Prep

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Combinatorics Difficulty 5.7 AIME, harder Find the answer Italy

Problem:

In the forest of the purple tree there are three types of animals capable of speaking: foxes, snakes and turtles. The first ones lie only on rainy days, the second ones always lie, the third ones always tell the truth. One day the explorer Berny talks with four animals. Their statements, reported in the order in which they were made, are:
A: "Today it is raining."
B: "The animal that spoke before me is lying."
C: "Today it is clear."
D: "The one who spoke before me is lying or I am a fox."
With how many turtles at most has Berny spoken?

Pick one

Solution

Solution:

The answer is (B). Let us consider the two possibilities: it is raining or it is not raining.

If it is raining, foxes and snakes lie, while turtles tell the truth. Then AA, who tells the truth, is necessarily a turtle. BB lies, so it is a fox or a snake. Likewise CC lies, so it is a fox or a snake. DD tells the truth, because CC lies, so it is a turtle. There are then exactly two turtles.

If it is not raining, snakes lie, while foxes and turtles tell the truth. Then AA, who lies, is necessarily a snake, while BB tells the truth and is therefore a fox or a turtle. CC tells the truth, so it too is a fox or a turtle. Since CC did not lie, DD can be a fox, in which case the statement is true, or a snake, in which case the statement is false. In this case there are therefore at most two turtles.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.