Question 38
Let $P_1(x)$ and $P_2(x)$ be monic quadratic polynomials (i.e., quadratic polynomials with leading coefficient $1$). Let points $A_1$ and $A_2$ be vertices of parabolas $y = P_1(x)$ and $y = P_2(x)$, respectively. By $m(g(x))$ denote the minimal value of the function $g(x)$. It happens that the differences $m(P_1(P_2(x))) - m(P_1(x))$ and $m(P_2(P_1(x))) - m(P_2(x))$ are equal positive real numbers. Find the angle between line $A_1A_2$ and the coordinate axis $Ox$. (Н. Х. Агаханов)