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April 2nd, 2013, 07:55 PM  #1 
Newbie Joined: Apr 2013 Posts: 6 Thanks: 0  Geometry(Triangle)/Algebra(Probability) Problem
A triangle will be called almost equilateral if the sum of the differences between all pairs of sides is 6 or less. Thus, a 345 triangle is almost equilateral, since 1 + 2 + 1 = 4. However, a 468 triangle is not, since 2 + 4 + 2 = 8. How many almost equilateral triangles are there with sides whose lengths are integers 100 or less, and such that no side length ends in the digits 4 or 6? (Note: a 345 triangle is considered the same as a a 354 triangle; the order of the sides does not matter.) Any and all help is much appreciated. Thank you! 
April 2nd, 2013, 07:57 PM  #2 
Newbie Joined: Apr 2013 Posts: 6 Thanks: 0  Re: Geometry(Triangle)/Algebra(Probability) Problem
I also have a variant on the same problem that I also must solve: How many almost equilateral triangles are there with sides whose lengths are integers 100 or less, and such that no side length ends in the digit 6? Once again, any and all help is much appreciated. Thank you and have a good day! 
April 2nd, 2013, 08:34 PM  #3  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 10,103 Thanks: 684  Re: Geometry(Triangle)/Algebra(Probability) Problem Quote:
1st 4: 122, 133, 144, 223 last 4: 9899100, 98100100, 9999100, 99100100 Call the sides abc a=1: 3 a=2: 6 a=3: 8 a=4 to 97: 9 a=98: 5 a=99: 2 a=100: 0 3 + 6 + 8 + 94*9 + 5 + 2 + 0 = 870 ba + ca + cb = 2c  2a 2c  2a = 6 c  a = 3 If c  a <= 3 then ok  

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