MathDB

An equilateral triangle

\Delta

of side length

L>0

is given. Suppose that

n

equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside

\Delta

, such that each unit equilateral triangle has sides parallel to

\Delta

, but with opposite orientation. (An example with

n=2

is drawn below.) [asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5)); filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); [/asy] Prove that

n \leq \frac{2}{3} L^{2}.

Problem Statement