System of equations with four unknown values

Question

I'm French so don't hesitate to tell me if I'm unclear.

It's been 3 hours I'm stuck with the following problem:

Find all the solutions $(a, b, c, d)$ of the system of equations where $a, b, c$ and $d$ are real numbers

\begin{cases} a+2023/a &=2b \\ b+2023/b &=2c \\ c+2023/c &=2d \\ d+2023/d &=2a \\ \end{cases}

What I've tried so far with my limited mathematical skills:

• Gaussian elimination (failure, it seems that this method can't apply with this type of system)
• Cramer's rule (again a failure)

Can someone give me a hint in order to resolve this problem? Thanks for your help.

Answer 1

Hints towards a solution:

• Note that $a, b, c, d$ have the same sign, so WLOG assume they are positive and we can apply $AM-GM$.
• Let $ a = \min (a, b, c, d)$. What can we say about $a$ from $ a + 2023 / a = 2b \geq 2a $?
• Show that $ b = a$.
• Hence conclude that $ a = b = c = d$.
• Thus, there are 2 solutions.