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Any two out of five (Posted on 2017-12-16) Difficulty: 5 of 5
Any two members out of (a1, a2, a3, a4, a5)
add up to a square number.

List the ten squares.

Rem: Existence of similar six-number set is not resolved yet.

See The Solution Submitted by Ady TZIDON    
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a quick note | Comment 3 of 5 |
A trivial solution has all the members = 2x^2 but I guess that's not what's wanted.

I use s1-s10 to represent the 10 squares and a=first member.

Then the desired set = (a, s1-a, s2-a, s3-a, s4-a).  The next three squares are given by the equations:

s1+s2-2a = s5
s1+s3-2a = s6
s1+s4-2a = s7 from which

s1-2a = s5-s2 = s6-s3 = s7-s4

So we're looking for integers that are the sum of two squares in two ways.  There are plenty of such squares, eg, 50,65,200,221.  The trick is to find such with factors that can be used in the other formulations.  Ideally all the squares would be distinct.     

  Posted by xdog on 2017-12-17 13:12:22
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