IMC 2002 · Day 1 · P1

First we show that the standard parabola with vertex $V$ contains point $A$ if and only if the standard parabola with vertex $s(A)$ contains point $s(V)$.

Let $A = (a, b)$ and $V = (v, w)$. The equation of the standard parabola with vertex $V = (v, w)$ is $y = (x - v)^2 + w$, so it contains point $A$ if and only if $b = (a - v)^2 + w$. Similarly, the equation of the parabola with vertex $s(A) = (a, -b)$ is $y = (x - a)^2 - b$; it contains point $s(V) = (v, -w)$ if and only if $-w = (v - a)^2 - b$. The two conditions are equivalent.

Now assume that the standard parabolas with vertices $V_1$ and $V_2$, $V_1$ and $V_3$, $V_2$ and $V_3$ intersect each other at points $A_3$, $A_2$, $A_1$, respectively. Then, by the statement above, the standard parabolas with vertices $s(A_1)$ and $s(A_2)$, $s(A_1)$ and $s(A_3)$, $s(A_2)$ and $s(A_3)$ intersect each other at points $V_3$, $V_2$, $V_1$, respectively, because they contain these points.