Determine the value of A
2 + B
2 + C
2 + D
2, 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
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.
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Posted by Charlie
on 2015-07-09 12:56:06 |
steps with some internet help
