3

Part of 1996 China Team Selection Test

Problems(2)

larger number is a multiple of the smaller number

Source: China TST 1996, problem 3

Let M \equal{} \lbrace 2, 3, 4, \ldots\, 1000 \rbrace. Find the smallest

n \in \mathbb{N}

n

-element subset of

M

contains 3 pairwise disjoint 4-element subsets

S, T, U

such that I. For any 2 elements in

S

, the larger number is a multiple of the smaller number. The same applies for

T

U

s \in S

t \in T

, (s,t) \equal{} 1. III. For any

s \in S

u \in U

(s,u) > 1

combinatorics unsolvedcombinatorics

|ka + lb + mc| > 1/h

Source: China TST 1996, problem 6

Does there exist non-zero complex numbers

a, b, c

and natural number

h

such that if integers

k, l, m

|k| + |l| + |m| \geq 1996

|ka + lb + mc| > \frac {1}{h}

complex numbersalgebra unsolvedalgebra