RO-NMO 2009 Problem 121

Find all functions $f\in\mathcal{C}^1 [0,1]$ that satisfy $f(1)=-1/6$ and $\int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx.$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.