Three fixed points in the plane are specified by their coordinates: A = (1,0), B = (0, 4), and C with y-coordinate equal to 0 and x-coordinate equal to the answer to last week’s Hat Trick problem divided by 1,440. Given a parabola in a plane, a point X is called interior to the parabola if any ray originating at X crosses the parabola at most once.
What is the area of the region of the plane consisting of the points X which are interior to all parabolas through A, B, and C?
Let A, B, and C be the three vertices of a triangle with side lengths 4, 5, and 6. Simone chooses a point D on the side (BC) opposite A, E on the side opposite B, and F on the side opposite C. None of D, E, or F coincide with A, B, or C. Simone then draws three circles, P through E, F, and A, Q through D, F, and B, and R through D, E, and C. She notices that each pair of these circles intersects in two points..
What is the minimum distance between an intersection point of one pair of these circles and an intersection point of another pair of these circles?