Oxford University MAT 2018 Solved Papers

Question : 5 of 10

Marks: +1, -0

A circle of radius 2, centred on the origin, is drawn on a grid of points with integer coordinates. Let

n

be the number of grid points that lie within or on the circle. What is the smallest amount the radius needs to increase by for there to be

2 n-5

grid points within or on the circle?

$√{5}-2$
$√{6}-2$
$√{8}-2$ ,
1,
(c) $√{8}$

Validate

Solution:

The circle of radius 2 contains 13 points (the point at

(1,1)

is inside, but the point at

(1,2)

is outside, as

√{2}< 2< √{5}

). So

n=13

and

2 n-5=21

. We want the smallest circle that contains 21 points, which is eight more than the previous circle. The next-closest points to the origin are the points at

(± 1, ± 2)

and

(± 2, ± 1)

of which there are eight. So the radius must increase by

√{5}-2

. The answer is (a).

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