# A Problem of Satisfying Inequalities

James Tanton has recently tweeted the following problem:

Is it always possible to replace the (10) X's in the expression

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Copyright © 1996-2018 Alexander Bogomolny
Is it always possible to replace the (10) X's in the expression

The answer is Yes, and the induction seems to be a suitable tool to tackle this problem. To see that, let's first agree to replace the inequality signs with a more generic symbol, say "?".

Second, observe, that the problem of replacing the X's with integers 1, 2, ..., n is equivalent to that of replacing the X's with members of an arbitrary increasing sequence _{1} < a_{2} < ... < a_{n}.

α_{1} ? α_{2} ? ... ? α_{n}

where α's are distinct members of {1, ..., n} then also

a_{α1} ? a_{α2} ? ... ? a_{αn},

and vice versa. This is so because, when the sequence {a_{k}} is increasing, the inequality _{m} < a_{n}.

The induction is on the number N of the inequality signs. The statement is obvious for

Assume it holds for k = m and let's show that it then holds for

The next step depends on whether the removed inequality sign is "<" or ">". In the former case all we have to do is to replace the last X with

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