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Problem 395

Number theory Difficulty 2.6 Multiple choice CEMC Fermat · Canada · 2016

A total of $n$ points are equally spaced around a circle and are labelled with the integers 1 to $n$ , in order. Two points are called diametrically opposite if the line segment joining them is a diameter of the circle. If the points labelled 7 and 35 are diametrically opposite, then $n$ equals

Pick one

Official solution

The number of points on the circle equals the number of spaces between the points around the circle.

Moving from the point labelled 7 to the point labelled 35 requires moving $35-7=28$ points and so 28 spaces around the circle.

Since the points labelled 7 and 35 are diametrically opposite, then moving along the circle from 7 to 35 results in travelling halfway around the circle.

Since 28 spaces makes half of the circle, then $2\cdot 28 = 56$ spaces make the whole circle.

Thus, there are 56 points on the circle, and so $n = 56$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.