MathDB

7

Part of 2000 IMO Shortlist

Problems(2)

At least how many gangsters will be killed?

Source: IMO Shortlist 2000, G7

8/10/2008
Ten gangsters are standing on a flat surface, and the distances between them are all distinct. At twelve o’clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed?
geometryPythagorean TheoremIMO Shortlist
Find all ints n for which n-independent polynomials exist

Source: IMO Shortlist 2000, A7

8/10/2008
For a polynomial P P of degree 2000 with distinct real coefficients let M(P) M(P) be the set of all polynomials that can be produced from P P by permutation of its coefficients. A polynomial P P will be called n n-independent if P(n) \equal{} 0 and we can get from any Q∈M(P) Q \in M(P) a polynomial Q1 Q_1 such that Q_1(n) \equal{} 0 by interchanging at most one pair of coefficients of Q. Q. Find all integers n n for which n n-independent polynomials exist.
algebrapolynomialmodular arithmeticcoefficientsIMO Shortlist