Not Too Tough
It is known that the numbers A1,A2,...............An and B1,B2,............Bn are both permutations of 1,1/2,........1/n.In addition,we know that A1+B1>=A2+B2............>=An+Bn.
If Am+Bm<=k/m. find minimum value of k.
(m,n,k belong to Natural set and n>10)
The answer is 4.
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1 solution
For every m(1<=m<=n) among the m pairs(Ax,Bx), 1 of the inequalities Ax>=Bx or Bx>Ax is satisfied at least in m/2 pairs.
For instance, let Bx>=Ax at least in m/2 pairs. If Bt is the smallest of these Bx, then Bt<=2/m. Hence At+Bt<=2Bt<=4/m, and since t<=m, we have
Am+Bm<=At+Bt<=4/m