# Using expected value to prove that there is a line intersecting at least 4 of the circles

There are several circles inside a square of side length $1$. The sum of the circumferences of the circles is $10$. Prove that there exists a line that intersects at least $4$ of the circles.

I know I have to use expected value to prove this, but how? Casework wouldn’t possibly work and I can’t really think of other stratigies. Thanks in advanced for providing a solution!

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Basic approach. Let the unit square be the square bounded by the four lines $x = 0$, $x = 1$, $y = 0$, and $y = 1$. Further let $f(u)$ be the number of circles that intersect the vertical line $x = u$. What is the expected value of $f(u)$ over the interval $[0, 1]$? In other words, what is $\int_{u=0}^1 f(u) \, du$? What can you say about the maximum of $f(u)$ over that same interval?

This is more or less a variation on the pigeonhole principle.