## PROBLEM 4

Let ABC be any triangle. Can this triangle be divided into 16384 congruent triangles? Justify your answer.
## Solution:

Let $M, N, P$ be the midpoints of sides $B C, C A$, and $A B$ of $\triangle A B C$ respectively.

We obtain $\Delta \boldsymbol{A P N} \equiv \Delta \boldsymbol{B P} \equiv \Delta \boldsymbol{C M N} \equiv \Delta \boldsymbol{P} \boldsymbol{M} \boldsymbol{N}$, according to the S.S.S. case. 2 points

By the same method, each of the 4 triangles will be divided into 4 more congruent triangles, resulting in $4^{2}$ congruent triangles. 1 point Each of the $4^{2}$ congruent triangles will then be divided into 4 more congruent triangles, resulting in $4^{3}$ congruent triangles. 1 point

The process continues until we obtain $4^{7}=16384$ congruent triangles...

2 points