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# Puzzles

## Find them all

Find all continuous positive functions, $$f$$ on $$[0,1]$$ such that:
$$\int_0^1 f(x) dx=1\\ \mathrm{and }\int_0^1 xf(x) dx=\alpha\\ \mathrm{and }\int_0^1 x^2f(x) dx=\alpha^2$$
If you enjoyed this puzzle, check out Sunday Afternoon Maths XXXIV,
puzzles about integration, or a random puzzle.

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