AMO-Bench 2025 Problem 1

Divide each side of an equilateral triangle $ABC$ into four equal parts. Through each division point, draw lines parallel to the other two sides. The sides of triangle$ABC$ together with these parallel lines define $15$ lattice points. Find the smallest integer $n$ such that, no matter which $n$ points are chosen among these $15$ lattice points, there always exist three points that form an isosceles triangle.