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June 15th, 2018, 11:00 PM   #1
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Another Complex Problem

I have come across another tough problem. Please help me to solve it. My thinking is $\displaystyle (1004)^2$ should be the first maximum value and the final maximum value is $\displaystyle (2007)^2$. But i don't know how to prove it will start from $\displaystyle (1004)^2$.
Attached Images Q6.jpg (7.9 KB, 19 views) June 17th, 2018, 04:05 AM   #2
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Quote:
 Originally Posted by MathsLearner123 I have come across another tough problem. Please help me to solve it. My thinking is $\displaystyle (1004)^2$ should be the first maximum value and the final maximum value is $\displaystyle (2007)^2$. But i don't know how to prove it will start from $\displaystyle (1004)^2$.
Why not look at the problem with 1-7 instead of 1-2007 and see what you can do? June 17th, 2018, 05:45 AM #3 Member   Joined: Aug 2017 From: India Posts: 54 Thanks: 2 Thanks for the hint. Yes I took from 1 to 7 and observed this If I assume the following set {a1,a2,a3,a4,a5,a6,a7} => {7,6,5,4,3,2,1} I end up as 1*7, 2*6, 3*5, 4*4, 5*3, 6*2, 7*1 => 7, 12, 15, 16, 15, 12, 7. Hence 16 is the greatest value obtained, can i generalize as? $\displaystyle ((1+7)/2)*((1+7)/2) = 4^2 = 16$ June 17th, 2018, 06:17 AM   #4
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 Originally Posted by MathsLearner123 Thanks for the hint. Yes I took from 1 to 7 and observed this If I assume the following set {a1,a2,a3,a4,a5,a6,a7} => {7,6,5,4,3,2,1} I end up as 1*7, 2*6, 3*5, 4*4, 5*3, 6*2, 7*1 => 7, 12, 15, 16, 15, 12, 7. Hence 16 is the greatest value obtained, can i generalize as? $\displaystyle ((1+7)/2)*((1+7)/2) = 4^2 = 16$
To generalise the result, try a proof by contradiction. Again, try first for 1-7. The question is what happens if you try to keep every product less than $4^2$. June 17th, 2018, 08:07 AM #5 Global Moderator   Joined: Dec 2006 Posts: 20,921 Thanks: 2203 For every product to be less than 1004², each of the 1004 integers from 1004 to 2007 must be paired with one of the 1003 integers from 1 to 2003, with no such number being used twice, which is impossible. Tags complex, problem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post noobinmath Algebra 8 November 1st, 2014 09:35 AM ZardoZ Algebra 9 January 23rd, 2013 11:44 PM buwwy Complex Analysis 8 January 1st, 2013 07:48 AM markrenton Complex Analysis 2 June 19th, 2010 06:00 AM

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