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China TST 1987 smallest integer from five sets

Source: China TST 1987, problem 1

May 16, 2005

combinatoricsNumber theoretic functionsSetsChinaTeam Selection Test

Problem Statement

a.) For all positive integer

k

find the smallest positive integer

f(k)

such that

5

sets

s_1,s_2, \ldots , s_5

exist satisfying: i. each has

k

elements; ii.

s_i

and

s_{i+1}

are disjoint for

i=1,2,...,5

(

s_6=s_1

) iii. the union of the

5

sets has exactly

f(k)

elements.
b.) Generalisation: Consider

n \geq 3

sets instead of

5