2012 Indonesia MO Problems/Problem 5
Problem
Given positive integers and
. Let
and
be two collections of
numbers of
and
, arranged in
rows and
columns. An example of such collections for
and
is
Let those two collections satisfy the following properties:
(i) On each row of
, from left to right, the numbers are non-increasing,
(ii) On each column of
, from top to bottom, the numbers are non-increasing,
(iii) The sum of numbers on the row in
equals to the same row in
,
(iv) The sum of numbers on the column in
equals to the same column in
.
Show that the number on row
and column
of
equals to the number on row
and column
of
for
and
.
Solution
let the collection X be named
by (i), for all
,
, that means
for the collection Q, all the 1's are getting pushed up and the inequality of the sum of columns of Q are pushing it to the left, if we do the same logic to P we also sed it getting pushed up and left. notice how if the sum of some row/column is some number
, then the first p numbers of the row/column is
and the rest
, so we can deduce its equal (sorry i have no idea how to tody the writing, some1 help :()
See Also
2012 Indonesia MO (Problems) | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 6 |
All Indonesia MO Problems and Solutions |