MathDB

3

Part of 1989 China Team Selection Test

Problems(2)

non-zero roots in the unitary circumference

Source: China TST 1989, problem 3

6/27/2005
Find the greatest nn such that (z+1)n=zn+1(z+1)^n = z^n + 1 has all its non-zero roots in the unitary circumference, e.g. (α+1)n=αn+1,α≠0(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0 implies ∣α∣=1.|\alpha| = 1.
Gausscalculusderivativealgebrapolynomialcombinatorial geometryChina
1989 equal circles are arbitrarily placed on the table

Source: China TST 1989, problem 7

6/27/2005
19891989 equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.
combinatorics unsolvedcombinatorics