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Distance between points of intersection of line and circle.

Source: ILL 1979-48

June 5, 2011

geometry unsolvedgeometry

Problem Statement

In the plane a circle

C

of unit radius is given. For any line

l

, a number

s(l)

is defined in the following way: If

l

and

C

intersect in two points,

s(l)

is their distance; otherwise,

s(l) = 0

. Let

P

be a point at distance

r

from the center of

C

. One defines

M(r)

to be the maximum value of the sum

s(m) + s(n)

, where

m

and

n

are variable mutually orthogonal lines through

P

. Determine the values of

r

for which

M(r) > 2

.

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