3

Part of 1998 China National Olympiad

Problems(2)

Least n such that any n element subset satisfies condition

Source: Chinese Mathematical Olympiad 1998 Problem 3

S=\{1,2,\ldots ,98\}

. Find the least natural number

n

such that we can pick out

10

n

-element subset of

S

satisfying the following condition: no matter how we equally divide the

10

numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.

number theory unsolvednumber theory

Maximal value of |x_k|

Source: Chinese Mathematical Olympiad 1998 Problem 6

x_1,x_2,\ldots ,x_n

be real numbers, where

n\ge 2

\sum_{i=1}^{n}x^2_i+ \sum_{i=1}^{n-1}x_ix_{i+1}=1

k

, find the maximal value of

|x_k|

inequalities unsolvedinequalities