2024 SSMO Relay Round 2 Problems/Problem 3

Problem

Let A point is randomly chosen inside the square with vertices and . Find the perimeter of the set containing all points for which

Solution

Let We have \begin{align*} \sqrt{x^2+y^2}+\sqrt{(X-T)^2+(y-T)^2}&\ge\sqrt{x^2+(y-T)^2}+\sqrt{(x-T)^2+y^2}\implies\\ x^2+y^2+(x-T)^2+(y-T^2)&+2\sqrt{x^2+y^2}\cdot\sqrt{(X-T)^2+(y-T)^2}\ge\\ x^2+(y-T)^2+(x-T)^2&+y^2+2\sqrt{x^2+(y-T)^2}\cdot\sqrt{(x-T)^2+y^2}\implies\\ (x^2+y^2)((x-T)^2+(y-T)^2)&\ge(x^2+(y-T)^2)(y^2+(x-T)^2)\implies\\ y^2(x-T)^2+(x^2)(y-T)^2&\ge x^2y^2+(x-T)^2(y-T)^2\implies\\ (x^2-(x-T)^2)(y^2-(y-T)^2)&\le0\implies\\ (T^2-2xT)(T^2-2yT)&\le0\implies\\ \left(x-\frac{T}{2}\right)\left(y-\frac{T}{2}\right)\le0. \end{align*} So, if we split the square into four smaller squares by drawing lines the set is equivalent to the bottom left and top right squares. Thus, the perimeter of the set is equivalent to

~SMO_Team