Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Find the answer

Example 1. Given f(1−x2)=1−x2x2f\left(1-x^{2}\right)=\frac{1-x^{2}}{x^{2}}. Find f(12)f\left(\frac{1}{2}\right).

A number or a short expression. Spacing and $ signs are ignored.

Solution

 Solve f(1−x2)=1−x21−(1−x2), hence f(x)=x1−x,f(12)=1.\begin{array}{l}\text { Solve } f\left(1-x^{2}\right)=\frac{1-x^{2}}{1-\left(1-x^{2}\right)}, \\ \text { hence } f(x)=\frac{x}{1-x}, f\left(\frac{1}{2}\right)=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.