satisfy that: 
,![\[
\sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a} \geq \frac{4(ab + bc + ca)}{\sqrt{(a + b)(b + c)(c + a)}}
\]](http://latex.artofproblemsolving.com/e/7/2/e72fd0f134289c359bc8766440babc52ae131961.png)
satisfied 
Find the minimum of 
Y by
Each of the tips of a hexagram, or regular six point star, is an equilateral triangle. One can construct another hexagram, using the midpoint of the internal base of these triangles as vertices. If we repeat this process once again, we obtain a sequence of three hexagrams as shown in the figure.
What is the sum of the areas of the tips of these hexagrams, if we know that the most external equilateral triangles have sides of length
?
What is the sum of the areas of the tips of these hexagrams, if we know that the most external equilateral triangles have sides of length
?This post has been edited 2 times. Last edited by parmenides51, Jul 22, 2021, 3:23 PM
equilateral triangular tips get halved. So the area gets quartered. Initial area is 
.![$$3*\sqrt{3}/2 * [1 +1/4 + ... ]$$](http://latex.artofproblemsolving.com/1/9/4/194672e094d22e96c8047d58b7b6f293c1f504d1.png)
.
