Beauty in Symmetry
⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ x + y + z + 1 8 a + b = 1 3 5 8 0 2 4 5 8 x + y + 1 8 z + a + b = 1 3 5 8 0 2 4 5 8 1 8 x + y + z + a + b = 1 3 5 8 0 2 4 5 8 x + 1 8 y + z + a + b = 1 3 5 8 0 2 4 5 8 x + y + z + a + 1 8 b = 1 3 5 8 0 2 4 5 8
Solve the system of equations above for real values of x , y , z , a , b .
Enter the value of x as your answer.
The answer is 6172839.
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1 solution
Yay! Same method!
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Thanks for your comment :) ... No matter how complex math we know and understand... These are the things I love the most.... How did you find it?
Same way dude!!!
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@abc xyz cool :) ... Thanks for your comment
Observing the symmetry in the equation, we see that interchanging the values of x , y , z , a , b cyclically in the first equation will yield b + x + y + 1 8 z + a = 1 3 5 8 0 2 4 5 8 which is the second equation. Similarly, interchanging the variables in cyclical manner would yield all the 5 equations in some order. This shows that all the variables have to be equal. putting x = y = z = a = b
2 2 x = 1 3 5 8 0 2 4 5 8 x = 6 1 7 2 8 3 9