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March 22nd, 2014, 07:26 PM  #1 
Senior Member Joined: Apr 2012 Posts: 799 Thanks: 1  which one is the lengthiest with side length : we have : now determine among : which one is the lengthiest 
March 22nd, 2014, 08:13 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra  Re: which one is the lengthiest
You want to express b in terms of a only and c in terms of a only. You can do this by eliminating b and c in turn from the equations in the same manner that you would to solve a system of simultaneous equations. You should then be able to order them appropriately. I made for all values of and when . 
March 22nd, 2014, 08:23 PM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  Re: which one is the lengthiest
c. Also, if a,b,c are integers, then b>a except for one case: (a,b,c) = (5,3,7) 
March 22nd, 2014, 09:26 PM  #4 
Senior Member Joined: Mar 2014 Posts: 112 Thanks: 8 
0 = aČ  4c + 3 i.e. 4c = aČ + 3. Hence c = (aČ + 3)/4. 0 = aČ  2a  4b  3 i.e. 4b = aČ  2a  3 = (a  3)(a + 1). Hence b = (a  3)(a + 1)/4. Since b > 0 is correct, a > 3 must also be correct. If we use the solution Denis gave we are done. 
March 24th, 2014, 09:14 PM  #5  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  Re: which one is the lengthiest Quote:
Above equation can be rewritten as: 2c  2b = a + 3 [1] So evident that c > b (since a,b,c are all > 0, being triangle sides) Can't find "as simple" a way to show c > a...  
March 25th, 2014, 11:24 AM  #6 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  Re: which one is the lengthiest
Albert, something for your "useless information(!)" file; here's the first 8 integer solutions: Code: n a b c 1 5 3 7 2 7 8 13 3 9 15 21 4 11 24 31 5 13 35 43 6 15 48 57 7 17 63 73 8 19 80 91 b = n(n + 2) c = n^2 + 3n + 3 You're welcome; no charge :P 