MathDB

6

Part of 2004 239 Open Mathematical Olympiad

Problems(2)

Subset in The Form of Union of Segments

Source: 239 2004 S P6

5/15/2020
Do there exist a set A⊂[0,1]A\subset [0,1] such that (a)(a) AA is a finite union of segments of total length 12\frac{1}{2}, (b)(b) The symmetric difference of AA and B:=A/2∪(A/2+1/2)B:=A/2\cup(A/2+1/2) is a union of segments of the total length less than 110000\frac{1}{10000}?
algebra
Divisibility of LCM

Source: 239 2004 J P6

5/15/2020
Given distinct positive integers a1, a2, …,ana_1,\,a_2,\,\dots,a_n. Let bi=(ai−a1)(ai−a2)…(ai−ai−1)(ai−ai+1)…(ai−an)b_i = (a_i - a_1) (a_i-a_2) \dots (a_i-a_{i-1}) (a_i-a_{i+1})\dots(a_i-a_n). Prove that the least common multiple [b1,b2,…,bn][b_1,b_2,\dots,b_n] is divisible by (n−1)!.(n-1)!.
number theoryleast common multiple