Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.1 AIME, harder Prove it Ukraine

Let $f(x) = x^2 - 8x + 20$ . Solve the equation $f(f(f(x))) = 5$ .

Solution

Неважко помітити, що $f(x) = (x - 4)^2 + 4$ . Тоді рівняння $f(f(f(x))) = 5$ рівносильне рівнянню $(x - 4)^8 + 4 = 5$ , звідки $x - 4 = \pm 1$ , тобто $x = 3$ або $x = 5$ .

Відповідь: $x = 3, x = 5$ .

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