Track / Stage 3 / 85 of 260 #565 of 2444

Problem 565

AMC 10/12, early questions

Geometry Difficulty 3.5 Find the answer South African Mathematics Olympiad · South Africa

Points $A_1$ , $A_2$ , $A_3$ ... are constructed as follows: the length $OA_1$ is $4$ , $\angle OA_1A_2 = 90^\circ$ and the length $A_1A_2 = 1$ ; then a right angle is constructed at $A_2$ to find $A_3$ , and so on as shown in the diagram.
The length of $OA_{21}$ is

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

6 By Pythagoras, $OA_2 = \sqrt{17}$ . Then $OA_3 = \sqrt{18}$ , $OA_4 = \sqrt{19}$ and so on, with $OA_n = \sqrt{n+15}$ , and thus $OA_{21} = \sqrt{36} = 6$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.