1968 IMO Problems/Problem 3

Problem

Consider the system of equations $ax_1^2 + bx_1 + c = x_2$ $ax_2^2 + bx_2 + c = x_3$ $\cdots$ $ax_{n-1}^2 + bx_{n-1} + c = x_n$ $ax_n^2 + bx_n + c = x_1$ with unknowns $x_1, x_2, \cdots, x_n$ where $a, b, c$ are real and $a \neq 0$ . Let $\Delta = (b - 1)^2 - 4ac$ . Prove that for this system

(a) if $\Delta < 0$ , there is no solution,

(b) if $\Delta = 0$ , there is exactly one solution,

(c) if $\Delta > 0$ , there is more than one solution.

Solution

Adding the $n$ equations together yields

$\sum_{i=1}^{n} (ax_i^2+(b-1)x_i+c)=0$

Let $s_i=ax_i^2+(b-1)x_i+c$ .

(a) If $\Delta<0$ , then there is no solution to the quadratic equation $ax^2+(b-1)x+c$ , as the determinant is negative. This implies that either $s_i>0$ for all $i$ , or $s_i<0$ for all $i$ . In either case the above summation cannot be 0, which implies that there are no solutions to the given system of equations. $\blacksquare$

(b) If $\Delta=0$ , then there is exactly one solution to the quadratic equation $ax^2+(b-1)x+c$ (let it be $r$ ), and either $s_i\geq 0$ for all $i$ , or $s_i\leq 0$ for all $i$ . The only way that the above summation is 0 is if $s_i=0$ for all $i$ . As there is exactly one $x_i$ that makes $s_i=0$ (namely $x_i=r$ ), then the only possible solution to the system of equations is $(x_1,x_2,\dots , x_n)=(r, r, \dots , r)$ . It's not hard to show that this works, so when $\Delta=0$ the system of equations has exactly one solution. $\blacksquare$

(c) If $\Delta>1$ , then there are exactly two solutions to the quadratic equation $ax^2+(b-1)x+c$ . Let the roots be $r_1$ and $r_2$ . If $x_i=r_1$ for all $i$ , then $ax_i^2+bx_i+c=ar_1^2+br_1+c=r_1=x_{i+1}$ . This shows that $(x_1,x_2,\dots ,x_n)=(r_1,r_1,\dots, r_1)$ is a solution. We can show that $(x_1,x_2,\dots ,x_n)=(r_2,r_2,\dots, r_2)$ is another solution using the same reasoning, which shows that the equation has more than one solution. $\blacksquare$

1968 IMO (Problems) • Resources
Preceded by Problem 2	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 4
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1968 IMO Problems/Problem 3

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