Let
be a finite sequence of real numbers. For each
, from the sequence
we construct a new sequence
in the following way.
1. We choose a partition
, where
and
are two disjoint sets, such that the expression
attains the smallest value. (We allow
or
to be empty; in this case the corresponding sum is 0.) If there are several such partitions, one is chosen arbitrarily.
2. We set
where
if
, and
if
.
Prove that for some
, the sequence
contains an element
such that
.
Author: Omid Hatami, Iran
![A_0 = (a_1,\dots,a_n)](/media/m/3/5/a/35ad8fb32caac1d177d6c1856ab6acde.png)
![k\geq 0](/media/m/8/6/a/86aacd6649e15d5a95d9fc77f9ed2d99.png)
![A_k = (x_1,\dots,x_k)](/media/m/6/8/9/689816a88bc55e0380633ea756eccbdf.png)
![A_{k + 1}](/media/m/3/c/6/3c6eddb07cd1184087466683b28c1002.png)
1. We choose a partition
![\{1,\dots,n\} = I\cup J](/media/m/b/1/c/b1c41e637c0e9bc24f69db9e17e555b3.png)
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
![J](/media/m/9/0/e/90ef5cc2558381e341da5808eb92126f.png)
![\left|\sum_{i\in I}x_i - \sum_{j\in J}x_j\right|](/media/m/e/c/f/ecf93635a5b9b3d0b4e4320f191a069f.png)
attains the smallest value. (We allow
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
![J](/media/m/9/0/e/90ef5cc2558381e341da5808eb92126f.png)
2. We set
![A_{k + 1} = (y_1,\dots,y_n)](/media/m/f/8/d/f8dc6265499717450b86475ced6cdb67.png)
![y_i = x_i + 1](/media/m/1/9/7/19728c734703b69295cbb8d391c6c2a1.png)
![i\in I](/media/m/a/8/d/a8dfc8a99a4dfd6cfa09fac92858b432.png)
![y_i = x_i - 1](/media/m/7/d/d/7ddc983fd89a1880b12f074ae570b0f7.png)
![i\in J](/media/m/5/b/6/5b61043a9f33364c245b9a94cc596d08.png)
Prove that for some
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![A_k](/media/m/b/2/e/b2e11cfbe70ea64468986bde48119d91.png)
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![|x|\geq\frac n2](/media/m/0/d/7/0d7c125489e41135cdc24728bc2c413d.png)
Author: Omid Hatami, Iran