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Power Sum Poser (Posted on 2014-12-16) Difficulty: 3 of 5
Given that each of A, B and C is a positive integer greater than 1 - find the total count of positive integers between 1925 and 2025 inclusively that can be expressed in the form AB+BC.

For how many of the above representations do A+B+C equal a perfect square?

No Solution Yet Submitted by K Sengupta    
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Solution computer solution | Comment 1 of 2
   For a = 2 To 2025
     For b = 2 To 13
       t1 = Int(a ^ b + 0.5)
       If t1 > 2025 Then Exit For
       For c = 2 To 13
         t2 = Int(b ^ c + 0.5)
         If t1 + t2 > 2025 Then Exit For
         If t1 + t2 >= 1925 And t1 + t2 <= 2025 Then
              Text1.Text = Text1.Text & mform(a, "###0") & mform(b, "###0") & mform(c, "###0")
              Text1.Text = Text1.Text & "    " & mform(t1, " ###0") & mform(t2, " ###0") & mform(t1 + t2, "     ###0")
              sq = a + b + c
              sr = Int(Sqr(sq) + 0.5)
              If sr * sr = sq Then Text1.Text = Text1.Text & Str(a) & Str(b) & Str(c) & "     " & Str(a + b + c)
              Text1.Text = Text1.Text & crlf
              DoEvents
         End If
         
       Next
     Next
   Next a

finds 13 representations of these numbers

   a   b   c      a^b  b^c   a^b + b^c
   2  10   3     1024 1000     2024
   3   6   4      729 1296     2025
  12   3   5     1728  243     1971
  31   2  10      961 1024     1985
  38   2   9     1444  512     1956          38 2 9      49
  41   2   8     1681  256     1937
  42   2   8     1764  256     2020
  43   2   7     1849  128     1977
  44   2   2     1936    4     1940
  44   2   3     1936    8     1944          44 2 3      49
  44   2   4     1936   16     1952
  44   2   5     1936   32     1968
  44   2   6     1936   64     2000
  
Of these, two formulations have a+b+c equal to a perfect square; these are marked with a repetition of a, b and c to the right, together with the total, which is in each case 49.  

  Posted by Charlie on 2014-12-16 15:22:16
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