MathDB

3

Part of 2013 China National Olympiad

Problems(2)

help: Find all positive real numbers

Source: 2013 China Mathematical Olympaid P3

1/12/2013
Find all positive real numbers tt with the following property: there exists an infinite set XX of real numbers such that the inequality max⁡{∣x−(a−d)∣,∣y−a∣,∣z−(a+d)∣}>td \max\{|x-(a-d)|,|y-a|,|z-(a+d)|\}>td holds for all (not necessarily distinct) x,y,z∈Xx,y,z\in X, all real numbers aa and all positive real numbers dd.
inequalitiesceiling functiongeometryinequalities proposed
Find the minimum positive integer

Source: 2013 China Mathematical Olympaid P6

1/14/2013
Let m,nm,n be positive integers. Find the minimum positive integer NN which satisfies the following condition. If there exists a set SS of integers that contains a complete residue system module mm such that ∣S∣=N| S | = N, then there exists a nonempty set A⊆SA \subseteq S so that n∣∑x∈Axn\mid {\sum\limits_{x \in A} x }.
abstract algebramodular arithmeticnumber theory proposednumber theory