IMO Shortlist 1968 problem 4
Dodao/la:
arhiva2. travnja 2012. Let
![a,b,c](/media/m/3/6/4/36454fdb50fc50f021324b33a6b513e3.png)
be real numbers with
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
non-zero. It is known that the real numbers
![x_1,x_2,\ldots,x_n](/media/m/3/a/4/3a498672e3108eccaf28c3c29f3916e2.png)
satisfy the
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
equations:
![\begin{align*}
ax_1^2+bx_1+c &= x_{2} \\
ax_2^2+bx_2 +c &= x_3 \\
\vdots \\
ax_n^2+bx_n+c &= x_1
\end{align*}](/media/m/1/8/1/181bf252f68a04f596cada2e9365dcfe.png)
Prove that the system has zero, one or more than one real solutions if
![(b-1)^2-4ac](/media/m/c/0/4/c04d9d0185707caf224eaf7c5492526f.png)
is negative, equal to zero or positive respectively.
%V0
Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations:$$$\begin{align*}
ax_1^2+bx_1+c &= x_{2} \\
ax_2^2+bx_2 +c &= x_3 \\
\vdots \\
ax_n^2+bx_n+c &= x_1
\end{align*}$$$
Prove that the system has zero, one or more than one real solutions if $(b-1)^2-4ac$ is negative, equal to zero or positive respectively.
Izvor: Međunarodna matematička olimpijada, shortlist 1968