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Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1986 Swedish Mathematical Competition
5
5
Part of
1986 Swedish Mathematical Competition
Problems
(1)
pn numbers in an array, fixed difference wanted
Source: 1986 Swedish Mathematical Competition p5
3/28/2021
In the arrangement of
p
n
pn
p
n
real numbers below, the difference between the greatest and smallest numbers in each row is at most
d
d
d
,
d
>
0
d > 0
d
>
0
.
a
11
a
12
.
.
.
a
1
n
a
21
a
22
.
.
.
a
2
n
.
.
.
.
.
.
.
.
.
a
n
1
a
n
2
.
.
.
a
n
n
\begin{array}{l} a_{11} \,\, a_{12} \,\, ... \,\, a_{1n}\\ a_{21} \,\, a_{22} \,\, ... \,\, a_{2n}\\ \,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\ \,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\ \,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\ a_{n1} \,\, a_{n2} \,\, ... \,\, a_{nn}\\ \end{array}
a
11
a
12
...
a
1
n
a
21
a
22
...
a
2
n
.
.
.
.
.
.
.
.
.
a
n
1
a
n
2
...
a
nn
Prove that, when the numbers in each column are rearranged in decreasing order, the difference between the greatest and smallest numbers in each row will still be at most d.
algebra
Sum