Question

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

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

Answer 1

The correct option is A on a circle

Suppose there exist points B and C such that no other point of S lies on C(A, B, C). We have $f\left(B\right)+f\left(C\right)\equiv 0\left(modp\right),$ so, if f(B)= $i\ne 0,$ then f(C) = P- i. Let $\sigma =\sum _{xϵs}f\left(X\right).$ 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 $\sigma +(b-1)i\equiv 0\left(modp\right),$ $\sigma +(c-1)(p-i)\equiv \left(modp\right),$ hence $b+-2\equiv 0\left(modp\right).$ Since $1\le b,c\le n-2,$ we have $2\le b+c\le 2n-4<p,$ 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.