Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME Find the answer

Define $a$ ? $=(a-1) /(a+1)$ for $a \neq-1$ . Determine all real values $N$ for which $(N ?)$ ?=\tan 15.

Your answer

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

Solution

Let $x=N$ ?. Then $(x-1) \cos 15=(x+1) \sin 15$ . Squaring and rearranging terms, and using the fact that $\cos ^{2} 15-\sin ^{2} 15=\cos 30=\frac{\sqrt{3}}{2}$ , we have $3 x^{2}-4 \sqrt{3} x+3=0$ . Solving, we find that $x=\sqrt{3}$ or \frac{\sqrt{3}}{3} $. However, we may reject the second root because it yields a negative value for$ (N ?) $?. Therefore$ x=\sqrt{3} $and$ N=\frac{1+x}{1-x}=\frac{1+\sqrt{3}}{1-\sqrt{3}}=-2-\sqrt{3}$.

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