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Floors are perfect squares

Source: ITAMO 2019 #4

May 3, 2019
algebraITAMO 2019

Problem Statement

Let ⌊x⌋\lfloor x \rfloor denote the greatest integer less than or equal to x.x.
Let λ≥1\lambda \geq 1 be a real number and nn be a positive integer with the property that ⌊λn+1⌋,⌊λn+2⌋,⋯ ,⌊λ4n⌋\lfloor \lambda^{n+1}\rfloor, \lfloor \lambda^{n+2}\rfloor ,\cdots, \lfloor \lambda^{4n}\rfloor are all perfect squares.. Prove that ⌊λ⌋\lfloor \lambda \rfloor is a perfect square..
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