Question 28
a) There are two sequences, each of $2003$ consecutive integers, and a board of $2$ rows and $2003$ columns.
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Decide if it is always possible to distribute the numbers of the first sequence in the first row and those of the second sequence in the second row, in such a way that the results obtained by adding the two numbers in each column form a new sequence of $2003$ consecutive numbers?
b) What if $2003$ is replaced by $2004$?
In both a) and b), if the answer is affirmative, explain how you would distribute the numbers, and if the answer is negative, justify why.
| | | | | ... | |
|---|---|---|---|-----|---|
| | | | | ... | |
Decide if it is always possible to distribute the numbers of the first sequence in the first row and those of the second sequence in the second row, in such a way that the results obtained by adding the two numbers in each column form a new sequence of $2003$ consecutive numbers?
b) What if $2003$ is replaced by $2004$?
In both a) and b), if the answer is affirmative, explain how you would distribute the numbers, and if the answer is negative, justify why.