On a covering problem for equilateral triangles (Q1010747)

Summary: Let \(T\) be a unit equilateral triangle, and \(T_1,\dots,T_n\) be \(n\) equilateral triangles that cover \(T\) and satisfy the following two conditions: (i) \(T_i\) has side length \(t_i (0<t_i<1)\); (ii) \(T_i\) is placed with each side parallel to a side of \(T\). We prove a conjecture of Zhang and Fan asserting that any covering that meets the above two conditions (i) and (ii) satisfies \(\sum_{i=1}^n t_i \geq 2\). We also show that this bound cannot be improved.