Maths Olympiad Prep

Library / /147 of 520

Algebra Difficulty 5.0 AIME Find the answer

6. For 0<x<10<x<1, if the complex number
z=x+isin⁡x z=\sqrt{x}+\mathrm{i} \sqrt{\sin x}

corresponds to a point, then the number of such points inside the unit circle is n=n=

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

Solution

6. 1 .

From the point on the unit circle, we have

x+\sin x=1(00(x \in(0,1)),whichmeans, which means \varphi(x)$ is a strictly increasing function.

Also, φ(0)=−10\varphi(0)=-10, so the equation x+sin⁡x=1x+\sin x=1 has exactly one real root in (0,1)(0,1).
Therefore, n=1n=1.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

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