In the interior of a unit square, there are $n(n\in \mathbb{N}^*)$ circles whose sum of areas is greater than $n-1$. Prove that the circles has at least a common point of intersection
I really don't know where to start. Thanks
In the interior of a unit square, there are $n(n\in \mathbb{N}^*)$ circles whose sum of areas is greater than $n-1$. Prove that the circles has at least a common point of intersection
I really don't know where to start. Thanks
EDIT after reinterpretation of the problem statement.
If all circles contain the center of the square, we are done. If a circle does not contain the center, then its diameter is $<\frac{\sqrt 2}2$ and its area is $<\frac\pi8$. For all other circles, we have that the diameter is $\le 1$ and area $\le\frac \pi 4$. Thus the total area $A$ of the circles is $n-1 By solving for $n-1$, we find $(n-1)<\frac{\frac\pi 8}{1-\frac\pi4}=1.829\ldots,$ i.e. $n\le 2$. The case $n=1$ is trivial. If $n=2$, the total area of the disks is $>1$ hence greater than the area of the unit square, hence they must intersect.