2018 USAJMO Problems/Problem 6

Problem 6

Karl starts with $n$ cards labeled $1,2,3,\dots,n$ lined up in a random order on his desk. He calls a pair $(a,b)$ of these cards swapped if $a>b$ and the card labeled $a$ is to the left of the card labeled $b$ . For instance, in the sequence of cards $3,1,4,2$ , there are three swapped pairs of cards, $(3,1)$ , $(3,2)$ , and $(4,2)$ .

He picks up the card labeled 1 and inserts it back into the sequence in the opposite position: if the card labeled 1 had $i$ card to its left, then it now has $i$ cards to its right. He then picks up the card labeled $2$ and reinserts it in the same manner, and so on until he has picked up and put back each of the cards $1,2,\dots,n$ exactly once in that order. (For example, the process starting at $3,1,4,2$ would be $3,1,4,2\to 3,4,1,2\to 2,3,4,1\to 2,4,3,1\to 2,3,4,1$ .)

Show that no matter what lineup of cards Karl started with, his final lineup has the same number of swapped pairs as the starting lineup.

Solution

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

2018 USAJMO (Problems • Resources)
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2018 USAJMO Problems/Problem 6

Problem 6

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