HMMT 二月 2011 · ALGCALC 赛 · 第 1 题

HMMT February 2011 — ALGCALC Round — Problem 1

Let a , b , and c be positive real numbers. Determine the largest total number of real roots that the 2 2 2 following three polynomials may have among them: ax + bx + c , bx + cx + a , and cx + ax + b .

解析

Let a , b , and c be positive real numbers. Determine the largest total number of real roots that the 2 2 2 following three polynomials may have among them: ax + bx + c , bx + cx + a , and cx + ax + b . 2 Answer: 4 If all the polynomials had real roots, their discriminants would all be nonnegative: a ≥ 2 2 2 2 4 bc, b ≥ 4 ca , and c ≥ 4 ab . Multiplying these inequalities gives ( abc ) ≥ 64( abc ) , a contradiction. Hence one of the quadratics has no real roots. The maximum of 4 real roots is attainable: for example, 2 1 2 the values ( a, b, c ) = (1 , 5 , 6) give − 2 , − 3 as roots to x + 5 x + 6 and − 1 , − as roots to 5 x + 6 x + 1. 5