Given any four positive, distinct, real numbers, show that one can
choose three numbers $A, B, C$ among them, such that all of the following
quadratic equations have only real roots or all of them have only
imaginary roots. $$Bx^2 + x + C = 0$$$$Cx^2 + x + A = 0$$$$Ax^2 + x + B = 0$$
We want to show that given 4 positive, distinct, real numbers, we can choose 3 such that $$1<4BC, 1<4CA, 1<4AB$$ or $$1>4BC, 1>4CA, 1>4AB$$
If all 4 given numbers are greater than $\dfrac{1}{2}$, then the first case will always be true irrespective of your choice. And if all 4 given numbers are lesser than $\dfrac{1}{2}$, then the second case will always be true irrespective of your choice. But I am not able to proceed after that.
Best Answer
Suppose we are given four real numbers $0\lt A \lt B \lt C \lt D $. If $1 \lt 4BC$, then $1\lt 4BC \lt 4BD \lt 4CD$; so it suffices to choose $B,C,D.$ If $1\ge 4BC$, then $1\ge 4BC \gt 4AC \gt 4AB$, so choose $A,B,C$ this time.