Maths Olympiad Prep

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Algebra Difficulty 4.4 AIME Prove it United States

Problem:
If xx and yy are two positive numbers less than 11, prove that
11−x2+11−y2≥21−xy \frac{1}{1-x^{2}} + \frac{1}{1-y^{2}} \geq \frac{2}{1-x y}

Solution

Solution:
First we use the inequality a+b≥2aba + b \geq 2 \sqrt{a b} and get
11−x2+11−y2≥2(1−x2)(1−y2) \frac{1}{1-x^{2}} + \frac{1}{1-y^{2}} \geq \frac{2}{\sqrt{(1-x^{2})(1-y^{2})}}
Now we notice that
(1−x2)(1−y2)=1+x2y2−x2−y2≤1+x2y2−2xy=(1−xy)2 (1-x^{2})(1-y^{2}) = 1 + x^{2} y^{2} - x^{2} - y^{2} \leq 1 + x^{2} y^{2} - 2 x y = (1-x y)^{2}
which implies that
2(1−x2)(1−y2)≥21−xy \frac{2}{\sqrt{(1-x^{2})(1-y^{2})}} \geq \frac{2}{1-x y}
and this completes the proof.

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