ISL 1979 Problem 70

There are $1979$ equilateral triangles: $T_1,T_2, . . . ,T_{1979}$ . A side of triangle $T_k$ is equal to $\frac{1}{k}$ , $k = 1,2, . . . ,1979$ . At what values of a number $a$ can one place all these triangles into the equilateral triangle with side length $a$ so that they don't intersect (points of contact are allowed)?