I have the following problem from USAMO 2005:
"Let $n$ be an integer greater than 1. Suppose $2n$ points are given in the plane, no three of which are collinear. Suppose $n$ of the given $2n$ points are colored blue and the other $n$ colored red. A line in the plane is called a balancing line if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side.
Prove that there exist at least two balancing lines."
I found a nice solution for this at this link:
"https://artofproblemsolving.com/wiki/index.php?title=2005_USAMO_Problems/Problem_5"
In this solution, he has defined a function $f(i)=b_i-r_i$ where $b_i$ and $r_i$ be the number of points encountered before $R_i$ where $R_i$ is the ith point encountered when rotating a random line passing through a point on the convex hull.. However the author claims that the function $f(i)$ does achieve the value zero at least once for every point on the convex hull. How can I prove this? The author has proven it but I fail to understand that proof, can't there be a straight jump from a value of 1 t a value of -1 for the function defined above?
Note that $R_i$ is the $i$th red point encountered by the line.
Consider $f(i) - f(i - 1) = (b_i - b_{i-1}) - (r_i - r_{i-1})$. Certainly $b_i - b_{i-1} \geq 0$, since any blue points encountered before $R_{i-1}$ were encountered before $R_i$. And $r_i = i - 1$ - since the red points encountered before $R_i$ are $\{R_1, \cdots, R_{i-1}\}$ - and so $r_i - r_{i-1} = 1$. Therefore $f(i) - f(i - 1) \geq -1$. So in particular, if $f(i-1) > 0$ then we cannot have $f(i) < 0$; put another way, if $f$ changes sign from positive to negative it must do so counting in $1$s, and therefore must be $0$ at some point. And since $f(1) \geq 0$ and $f(n) \leq 0$, $f$ does change sign from positive to negative, so it must be $0$ at some point.
Btw - the linked proof isn't very well written, with several minor errors, but the core of the argument is sound.