IMO 1971 SL 8

Origin: NET

Problem

Determine whether there exist distinct real numbers a, b, c, t for which: (i) the equation ax2 + btx + c = 0 has two distinct real roots x1, x2, (ii) the equation bx2 + ctx + a = 0 has two distinct real roots x2, x3, (iii) the equation cx2 + atx + b = 0 has two distinct real roots x3, x1.

Solution

Suppose that a, b, c, t satisfy all the conditions. Then abc ̸= 0 and x1x2 = c a, x2x3 = a b , x3x1 = b c. Multiplying these equations, we obtain x2 1x2 2x2 3 = 1, and hence x1x2x3 = \epsilon = \pm1. From (1) we get x1 = \epsilonb/a, x2 = \epsilonc/b, x3 = \epsilona/c. Substituting x1 in the first equation, we get ab2/a2 + t\epsilonb2/a + c = 0, which gives us b2(1 + t\epsilon) = −ac. (1) Analogously, c2(1 + t\epsilon) = −ab and a2(1 + t\epsilon) = −bc, and therefore (1 + t\epsilon)3 = −1; i.e., 1 + t\epsilon = −1, since it is real. This also implies together with (1) that b2 = ac, c2 = ab, and a2 = bc, and consequently a = b = c. Thus the three equations in the problem are equal, which is impossible. Hence, such a, b, c, t do not exist.