Team Selection Test for IMO 2011

Let $p=2011$, $A=\{(i,j):0\le i,j<p\}$ and $B=\{k:0\le k<p-1\}$. We define $f$ as follows: $$f((i_1,j_1),(i_2,j_2))=\{\begin{array}{ll}\frac{j_1-j_2}{i_1-i_2}(\mathrm{mod}p)& \text{if }{i}_1\ne i_2\text{ and }\frac{j_1-j_2}{i_1-i_2}\ne -1(\mathrm{mod}p)\\ 0& \text{otherwise}\end{array}$$ In other words, we send a pair of points to the slope of the line passing through them unless they lie on a line with slope $\infty$ or $-1$, in which case we send it to $0$.

Let $g:A\to B$ be a function. Since $\frac{|A|}{|B|}=\frac{p^2}{p-1}>p$, there is a value in $B$ that is taken at least $p+1$ times by $g$. Since there are exactly $p$ lines with slope equal to this value, there is at least a pair of points lying on the same one of these lines.