CN-TST 2023 Problem 5

Let $\triangle ABC$ be a triangle, and let $P_1,\cdots,P_n$ be points inside where no three given points are collinear. Prove that we can partition $\triangle ABC$ into $2n+1$ triangles such that their vertices are among $A,B,C,P_1,\cdots,P_n$ , and at least $n+\sqrt{n}+1$ of them contain at least one of $A,B,C$ .