13th USAMO 1984 Problem 1 Two roots of the real quartic x 4 - 18x 3 + ax 2 + 200x - 1984 = 0 have product -32. Find a.
Let the two roots satisfy the quadratic x 2 + hx -32 = 0 (we have not yet shown that h is real). The other two roots have product -1984/-32 = 62. Let them satisfy the quadratic x 2 + kx + 62. So x 4 - 18x 3 + ax 2 + 200x - 1984 = (x 2 + hx -32)(x 2 + kx + 62) = x 4 + (h+k) x 3 + (hk + 30)x 2 + (62h - 32k)x - 1984 = 0. Equating coefficients: h + k = -18, 62h - 32k = 200. Solving, h = -4, k = -14. Hence a = 86. 13th USAMO 1984 © John Scholes jscholes@kalva.demon.co.uk 24 Aug 2002