Find all quadruplets of real numbers $(a, b, c, d)$ such that

$$
a=b c+c d, b=c d+d a, c=d a+a b \text{ and } d=a b+b c
$$
Let $(a, b, c, d)$ be a potential solution. The first step is to factorize the right-hand sides of our equations:

$$
a=c \times(b+d), b=d \times(a+c), c=a \times(b+d) \text { and } d=b \times(a+c) \text {. }
$$

Now, if $a=0$, the statement indicates that $c=0 \times(b+d)=0$, then $b=d \times 0=0$ and $d=b \times 0=0$. Furthermore, since the four numbers play cyclic roles, any other potential solution consists of four non-zero real numbers.
Under these new conditions, since $a c=a c \times(b+d)^{2} \neq 0$, we know that $b+d= \pm 1$. Similarly, $a+c= \pm 1$. Since $a+c=(a+c) \times(b+d)$, we deduce that $b+d=1$ and, similarly, that $a+c=1$. From then on, the equations in the statement can be rewritten as $a=c$ and $b=d$, so that $a=b=c=d=1 / 2$.
In conclusion, the only two potential solutions are $(0,0,0,0)$ and $(1 / 2,1 / 2,1 / 2,1 / 2)$, and it is easily verified that these are indeed solutions to the problem.

Comment from the graders: The problem was approached by the vast majority of students, but was not very well solved, as more than half of the papers contain a serious error among the following two:
$\triangleright$ Many students divide equations by a quantity (such as $a$ or $b+d$ most often) without checking if it is non-zero, which explains the absence of the solution $(0,0,0,0)$ in too many papers. Dividing by 0 is one of the most common errors in all algebra and arithmetic papers, and is yet easily avoidable: it must be remedied!
$\triangleright$ Several students attempt to use inequalities, but quickly find themselves multiplying by $b+d$ without checking the sign of this quantity. Multiplying by a negative number changes the direction of the inequality, so the entire proof falls apart from there. It was possible to reason with inequalities, but this was very laborious, and only one paper did it perfectly.
Finally, too many students forget to verify their solutions. Here, the verification is easy to do, but at least a mention that it has been done is necessary to show that this step has not been neglected. A sentence like "and we verify conversely that these solutions work" is the minimum necessary to show that this step has not been overlooked.