A math problem of HMMT general part to GeorgeLet a, b, and c be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: ax^2 + bx + c, bx^2 + cx + a, and cx^2 + ax + b.
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![A math problem of HMMT general part to GeorgeLet a, b, and c be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: ax^2 + bx + c, bx^2 + cx + a, and cx^2 + ax + b.](/uploads/image/z/14024739-3-9.jpg?t=A+math+problem+of+HMMT+general+part+to+GeorgeLet+a%2C+b%2C+and+c+be+positive+real+numbers.+Determine+the+largest+total+number+of+real+roots+that+the+following+three+polynomials+may+have+among+them%3A+ax%5E2+%2B+bx+%2B+c%2C+bx%5E2+%2B+cx+%2B+a%2C+and+cx%5E2+%2B+ax+%2B+b.)
A math problem of HMMT general part to GeorgeLet a, b, and c be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: ax^2 + bx + c, bx^2 + cx + a, and cx^2 + ax + b.
A math problem of HMMT general part to George
Let a, b, and c be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: ax^2 + bx + c, bx^2 + cx + a, and cx^2 + ax + b.
A math problem of HMMT general part to GeorgeLet a, b, and c be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: ax^2 + bx + c, bx^2 + cx + a, and cx^2 + ax + b.
If all the polynomials had real roots,their discriminants would all be nonnegative:a^2>=4bc; b^2>=4ca,and c^2>=4ab.Multiplying these inequalities gives (abc)^2>=64(abc)^2,a contradiction.
Hence one of the quadratics has no real roots.The maximum of 4 real roots is attainable:for example,
the values (a; b; c) = (1; 5; 6) give -2;-3 as roots to x^2 +5x+6 and -1/2,-1/3 as roots to 6x^2+5x+1.