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Question

# Let n≥4 be an integer and let p≥2n−3 be a prime number. Let S be a set of n points in the plane, no three of which are collinear, and let f:S→{0,1,...,p−1} be a function such that

A
on a circle
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B
in a circle
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C
outside the circle
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D
None of the above
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Solution

## The correct option is A on a circleSuppose there exist points B and C such that no other point of S lies on C(A, B, C). We have f(B)+f(C)≡0(modp), so, if f(B)= i≠0, then f(C) = P- i. Let σ=∑xϵsf(X). If a number of b circles pass through A, B and other points of S and a number of c circle pass through A, C and other points of S, applying the condition from the hypothesis to all these circles, we obtain σ+(b−1)i≡0(modp), σ+(c−1)(p−i)≡(modp), hence b+−2≡0(modp). Since 1≤b,c≤n−2, we have 2≤b+c≤2n−4<p, hence b= c= 1, which is a contradiction. It follows that for any points B, C ϵ S, there exist at least one more point of S lying on C(A, B, C). This implies the fact that all the points of S lie on a circle.

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