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x^2 + y^2 > 1 if (x^2 + y^2 -1)(z^2 + t^2 - 1) > (xz + yt -1)^2

Source: 2018 Grand Duchy of Lithuania, Mathematical Contest p1 (Baltic Way TST)

October 3, 2020
algebrainequalities

Problem Statement

Let x,y,z,tx, y, z, t be real numbers such that (x2+y2−1)(z2+t2−1)>(xz+yt−1)2(x^2 + y^2 -1)(z^2 + t^2 - 1) > (xz + yt -1)^2. Prove that x2+y2>1x^2 + y^2 > 1.
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