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February 10th, 2018, 04:08 AM  #1 
Newbie Joined: Dec 2017 From: Spain Posts: 13 Thanks: 0  Problem with inequalities
Good Morning, I've recently been working on a problem, which I haven't been able to solve: "Find the biggest positive integer with the following property: each digit except for the first and the last one must be smaller than the arithmetic mean of the two immediately adjacent digits. The veracity of the result must be proved." So for instance for a number [abc] b<((a+c)/2) I've found out that the biggest natural number with this condition has to be 96433469. My maths teacher told me that the result was the right one. He also said that I now only have to prove that there is no 9digit number that satisfies this condition. Does anybody have an idea how to solve this? Thanks Last edited by skipjack; February 12th, 2018 at 03:05 PM. 
February 12th, 2018, 03:42 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,574 Thanks: 1485 
Consider the signed differences between consecutive digits. For your 9digit value, these are 9  6 = 3, 6  4 = 2, 4  3 = 1, 3  3 = 0, 3  4 = 1, 4  6 = 2, and 6  9 = 3. These differences must form a strictly descending sequence. For a 10digit solution or a larger 9digit solution, what could that sequence be? Last edited by skipjack; February 16th, 2018 at 06:55 AM. 

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