Russion Problem 27 (1998-09-15)
Given 5 circles. Every 4 have a common point.
Prove there is a point on all 5.
SOLUTION
Let the circles be a, b, c, d and e. Let A be a point common to b, c, d,
e, let B be a point common to a, c, d, e and so on. If any two of A, B,
C, D, E coincide then the coincident point is on all 5 circles. Suppose
they are all distinct. Then A, B, C are on d and e. hence d and e
coincide (3 points determine a circle). Hence D is on all 5 circles.
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John Scholes
Note: This is a Helly property