IMO 1998 / 2

[Phlembac Adib Hasan]

In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that \[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]

Hint:

Follow the routine method i.e. counting the elements of a special set in two ways. In this case, \[A=\{(a_h,b_i,c_j)| a_h,b_i\text{ agree on candidate } c_j\}\]

And (if you are stuck) use this:

For odd $n$,\[\binom{x}{2}+\binom{n-x}{2}\ge\frac{(n-1)^2}{4},0\leq x\leq n\]