Discs in Triangle

First, let the centers of the circles with radius $2,3,4$ be $O_a,O_b,O_c$ respectively. Also, let the vertices of the equilateral triangle be $A,B,C$ . Suppose that circle $O_c$ is tangent to $AC$ and $BC$ ; and circle $O_b$ is tangent to $AB$ and $BC$ .

Drawing the line of symmetry $\overline{CO_c}$ , we see that if we can fit the circle $O_b$ at the place it is, we can fit a similar circle that is tangent to $AC$ and $AB$ , with the same radius. Therefore, the actual $O_a$ with radius $2$ , can fit in the remaining space with no question; we need only to consider $O_b$ and $O_c$ .

To minimize the space taken, we let $O_b$ and $O_c$ be tangent. In addition, let the points of tangency of $O_b$ , $O_c$ to $BC$ be $X$ , $Y$ , respectively. We can see from $30-60-90$ triangles that $CY=4\sqrt{3}$ and $BX=3\sqrt{3}$ . Also, $XY=\sqrt{(4+3)^2-(4-3)^2}=4\sqrt{3}$ . Therefore, $BC=11\sqrt{3}\approx \boxed{19.05}$ , as desired.

As a side note, I don't really think that this problem should be in this set... Perhaps the Level 4 set would have been more appropriate, with the frustum cup filled with water problem replacing this in level 5.

add the diameter all of the circles, we will find 18, so the smallest side length of the equilateral triangle approximately 19

tan 60° ×(4+3)+4+3 = ans