Italy | a nice Math Olympiad Question

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[@VictorPensioner]

Raise both sides of the equation to the sixth power
(1 + √(1+x))³ = x²
Let
у = √(1+x) => x = y² - 1
Then
(1 + y)³ = (y² - 1)² = (y - 1)² * (y + 1)²
or
y + 1 = 0 => x = 0
(1 + y) = (y - 1)² => y² - 3y = 0 => x = -1, x = 8
remains after verifying
x = 8