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Real Number Crossed Symmetric System of Equations (Posted on 2023-09-18) Difficulty: 3 of 5
Each of x, y, and z is a real number that satisfies this system of equations:
• 1/x+y+z=3
• x+1/y+z=3
• x+y+1/z=3
Determine all possible solutions to the above system of equations.

Note: Adapted from a problem that appeared on 2017 El Salvador Math Olympiad.

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Solution solution | Comment 1 of 2
I get 5 solutions (3 of which are the same except for swithing the order of x,y,z

By inspection, (1,1,1) is a solution

Case one:  Suppose x=y=z; 
1/x+2x=3; 2x^2 - 3x + 1 = 0
(x-1)(2x-1)=0; so (1/2,1/2,1/2) is also a solution

Case two:  Suppose 2 are equal but the third is different
wlog,  1/x+x+z=3
1/x+x+z=3
x+1/x+z=3
x+x+1/z=3
x+1/x = 2x; 1/x = x; x = ±1; -1 does not solve 3rd eqn.
eqn 3:  1/z = 3-2x; z=1/(3-2x); also z = 3-x-1/x
1/(3-2x) = 3-x-1/x
1 = (3-2x) (3-x-1/x)
1 = 9 - 3x - 3/x - 6x + 2x^2 + 2
2x^3 - 9x^2 + 10x - 3 = 0
(x-1)(2x^2 - 7x + 3) = 0
det: 49 - 24 = 25
quadratic's roots are (7 ± 5) / 4 = {3, 1/2}
The other root is x=y=1, we already have (1,1,1)
The last root is x=y=1/2, we already have (1/2,1/2,1/2)
try x=y=3
1/3 + 3 + (-1/3) = 3
2*3 + (-3) = 3       (3,3,-1/3)(3,-1/3,3)(-1/3,3,3) are solutions

Case three:  Suppose all different
Add 3 equations
(1/x) + (1/y) + (1/z) + 2(x+y+z) = 9
not sure this helps

3-z = 1/x+y = x+1/y
y + xy^2 = x^2y + x
y + xy^2 - x^2y - x = 0
(y-x) + xy(y-x) = 0
(y-x)(1+xy) = 0  note (y-x) not zero, since assuming all different

(1+xy) = 0  x = -1/y
(1+xz) = 0  x = -1/z
(1+yz) = 0
Therefore y=z which is a contradiction.
So no solutions for case three.

Summary, I get 5 solutions for (x,y,z):
(1,1,1)
(1/2,1/2,1/2)
(3,3,-1/3)
(3,-1/3,3)
(-1/3,3,3)

  Posted by Larry on 2023-09-18 09:58:32
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