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Oh, a China Math Olympiad problem! These are always a good challenge. I like that they're starting with the piecewise definition of the function and immediately highlighting the condition that f(A) = f(B) = f(C). That sets up the core idea right away.

That's a crucial point – A, B, and C must come from different segments because the function has to be distinct for each. The mention of 'turning points' at X=1 and X=9 is key, as that's where the function's behavior changes, which is critical for analyzing distinct inputs yielding the same output.

So, the goal is to find the range of ABC, and they've already established ABC = 9C. This means all our focus needs to shift to finding the constraints on C. That's a smart simplification to make the problem more manageable.

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The problem begins by introducing a piecewise function defined for X greater than zero [0:00]. For values of X less than 9, the function is |log base 3(X) - 1|, and for values greater than or equal to 9, it is 4 - sqrt(X). The core challenge lies in understanding that if three distinct inputs A, B, and C produce the same function output, they must each originate from different segments of the piecewise definition [0:20]. This implies a need to analyze the behavior and "turning points" of the function, such as where 3X equals 3 (resulting in X=1) and X=9, which are crucial for determining if the function is strictly decreasing or increasing.

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