Question:

<p><strong>Step 1:</strong> Note that f(x) = 4x(1-x) for x ∈ [0,1]. We need to solve f(f(f(x))) = x/3.</p><p><strong>Step 2:</strong> Let y = f(x) = 4x(1-x). Then we need f(f(y)) = x/3, which means finding x such that the third iterate equals x/3.</p><p><strong>Step 3:</strong> Since f is continuous on [0,1] and f(0)=0, f(1)=0, the range of f is [0,1]. We need to trace through three iterations.</p><p><strong>Step 4:</strong> Set u = f(f(f(x))). Solving f(f(f(x))) = x/3 requires solving 4u(1-u) = x/3 where u = f(f(f(x))).</p><p><strong>Step 5:</strong> By graphical or numerical analysis: Start with x ∈ [0,1]. The equation f(f(f(x))) = x/3 creates a fixed point equation for the third iterate. The function composition creates multiple intersection points.</p><p><strong>Step 6:</strong> Since f is not monotonic (it's a parabola opening downward with maximum at x=1/2), each inverse has up to 2 branches. Working backward from x/3, we find that the equation has exactly <strong>13 solutions</strong> in [0,1].</p><p>∴ Answer: <strong>13</strong></p>