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#1 2008-09-13 12:16:21

tony123
Member
Registered: 2007-08-03
Posts: 189

How many

How many triangles are there with integer sides and perimeter 2007?  How many of these are equilateral?  How many are isosceles?  How many are scalene

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#2 2008-09-14 08:22:54

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: How many

Let the triangle have sides a,b,c, with a≥b≥c. We want to solve the equation a+b+c=2007 in positive integers.
we have first that 669≤a≤1003, because if a is less that 2007/3=669, then b or c must be greater than a, if a is larger than 1003, then a>b+c contradicting the triangle inequality.
for a=669, then b=c=669 we have the equilateral triangle. Assume that a is greater than 669. If a is odd, then we start with the triangle b=c=(2007-a)/2. Now we may increase b and decrease c up to b=a, ie for any odd a, there are a-(2007-a)/2+1=3a/2-2005/2 triangles. For even a, we start with b=(2008-a)/2, c=(2006-a)/2. Now there are a-b+1=a-(2008-a)/2+1=(3a/2-2005/2)-1/2  different triangles. Summing these two yields number of triangles to:


the 167/2 is the -1/2 i pulled out from all the even numbers in the summation, altough im a bit tired so I may have calculated wrong somewhere

Last edited by Kurre (2008-09-14 08:45:26)

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