3

Part of 2004 France Team Selection Test

Problems(2)

Same as German TST

Source: French TST 2004 pb.3

Each point of the plane with two integer coordinates is the center of a disk with radius

\frac {1} {1000}

. Prove that there exists an equilateral triangle whose vertices belong to distinct disks. Prove that such a triangle has side-length greater than 96.

analytic geometryinequalitiespigeonhole principletriangle inequalitycomplex numbersgeometry proposedgeometry

Best wishes from Romania

Source: French TST 2004, pb.6

P

be the set of prime numbers. Consider a subset

M

P

with at least three elements. We assume that, for each non empty and finite subset

A

M

A \neq M

, the prime divisors of the integer

( \prod_{p \in A} ) - 1

M

M = P

inductionnumber theoryprime numbersnumber theory solved