IMO Shortlist 1968 problem 4


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2. travnja 2012.
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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