Problem

Let a, b, c be real numbers, a\ne 0. Prove that the system of equations

ax^{2}_{1}+bx_{1}+c=x_{2}
ax^{2}_{2}+bx_{2}+c=x_{3}
\cdots\cdots
ax^{2}_{n-1}+bx_{n-1}+c=x_{n}
ax^{2}_{n}+bx_{n}+c=x_{1}

(a) has no real solutions if (b-1)^{2}-4ac\lt 0 ;

(b) has a unique real solution if (b-1)^{2}-4ac=0 ;

(c) has more than one real solution if (b-1)^{2}-4ac\gt 0 .

Solution

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