Maths Olympiad Prep

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Algebra Difficulty 5.8 AIME, harder Prove it

Example 6 Given x≤1,y≤1x \leq 1, y \leq 1, prove that xy±(1−x2)(1−y2)‾≤1x y \pm \overline{\left(1-x^{2}\right)\left(1-y^{2}\right)} \leq 1.

Solution

 Prove: xy±(1−x2)(1−y2)‾\begin{array}{l} \text { Prove: } x y \pm \overline{\left(1-x^{2}\right)\left(1-y^{2}\right)} \end{array}
=1\begin{array}{l} =1 \end{array}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.