Question 157
Let $A$ be one of two points of intersection of two unequal circles in the same plane. If two particles from $A$ move each on a circle in a clockwise direction with two uniform velocities until they return to point $A$ at the same instant. Prove that there is always a point in the plane that is equidistant from the two particles at any moment while they are in motion.
Problem:
Let the parabola $y = x^2 + px + q$ be given, which intersects coordinate axes in 3 different points. Consider the circumcircle of the triangle having vertices of these 3 points. Prove that there is a point that belongs to that circle, regardless of values $p$ and $q$. Find that point.
Problem:
Let the parabola $y = x^2 + px + q$ be given, which intersects coordinate axes in 3 different points. Consider the circumcircle of the triangle having vertices of these 3 points. Prove that there is a point that belongs to that circle, regardless of values $p$ and $q$. Find that point.