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Polish MO Final 2010, 2nd problem (equation in rationals)

Source:

November 7, 2010
functionalgebrapolynomialnumber theory proposednumber theory

Problem Statement

Positive rational number aa and bb satisfy the equality a3+4a2b=4a2+b4.a^3 + 4a^2b = 4a^2 + b^4. Prove that the number a−1\sqrt{a}-1 is a square of a rational number.
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