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Inverse Quadratic Sum Puzzle (Posted on 2015-07-09) Difficulty: 3 of 5
Determine the value of A2 + B2 + C2 + D2, given that the following set of four equations are satisfied simultaneously:
  
   A2       B2       C2       D2
 ------ + ------ + ------ + ------ = 1
  22-1    22-32    22-52     22-72

   A2       B2       C2       D2
 ------ + ------ + ------ + ------ = 1
  42-1    42-32    42-52     42-72

   A2       B2       C2       D2
 ------ + ------ + ------ + ------ = 1
  62-1    62-32    62-52     62-72
   A2       B2       C2       D2
 ------ + ------ + ------ + ------ = 1
  82-1    82-32    82-52     82-72

No Solution Yet Submitted by K Sengupta    
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Solution steps with some internet help Comment 1 of 1
Using lower case a to represent A^2, etc.:

Starting:

a/3 - b/5 - c/21 - d/45 = 1
a/15 + b/7 - c/9 - d/33 = 1
a/35 + b/27 + c/11 - d/13 = 1
a/63 + b/55 + c/39 + d/15 = 1

Then:

105 a - 63 b - 15 c - 7 d = 315
231 a + 495 b - 385 c - 105 d = 3465
3861 a + 5005 b + 12285 c - 10395 d = 135135
715 a + 819 b + 1155 c + 3003 d = 45045


a = 11025/1024 = (105/32)^2 = A^2
b = 10395/1024
c =  9009/1024
d =  6435/1024

To get the sum:

a+b+c+d = A^2 + B^2 + C^2 + D^2 = 36

from

   10   A=11025//1024
   20   B=10395//1024
   30   C=9009//1024
   40   D=6435//1024
   60   print A//3-B//5-C//21-D//45
   70   print A//15+B//7-C//9-D//33
   80   print A//35+B//27+C//11-D//13
   90   print A//63+B//55+C//39+D//15
  100   print A+B+C+D

where the capital letters (imposed by UBASIC) represent my lower-case letters. Lines 60-90 verify the original set of equations.

  Posted by Charlie on 2015-07-09 12:56:06
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