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Problem 401

Combinatorics Difficulty 2.0 Prove it UKMT Junior Mathematical Olympiad · United Kingdom · 2017

A 3-digit integer is called a 'V-number' if the digits go 'high-low-high' – that is, if the tens digit is smaller than both the hundreds digit and the units (or 'ones') digit. How many 3-digit 'V-numbers' are there?

This one wants a proof. Work it on paper, then check yourself against the publisher's own solution, linked below. Be honest about it: the record is only any use to you if it is.

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Source: UK Mathematics Trust, licensed © UK Mathematics Trust; past papers published free at ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project. Solutions are the publisher's, linked not copied.