France | A Nice Number Theory Problem | Math Olympiad

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[@ToanPham-wr7xe]

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[@anthony17mapoy46]

Wrong! It should be "No Real Solution."
For complex solutions, we get x = -(y(1±i√3))/2.
In other words, x = -(y(1+i√3))/2 and x = -(y(1-i√3))/2.

[@shmuelzehavi4940]

You're absolutely right. No one has mentioned that x,y ∈ R.
But in the case of complex numbers we have to add also that: x, y, x+y ≠ 0.

[@renesperb]

This problem has infinitely many solutions: set z= 1/2 (-1 +i√3 ) . Choose any x (not zero) ,and set y = x*z.Then ,
1/x+1/y = 1/(x+y) . The same is true if you choose z =1/2 (-1 - i√3 )

[@martingibbs8972]

Did they really have a question like this in the Olympiad without stating whether x and y could be real or complex? That’s really poor.