The following is question A1 from the 2002 IMO:
$S$ is the set of all $(h,k)$ with $h,k$ non-negative integers such that $h+k<n$. Each element of $S$ is colored red or blue such that if $(h,k)$ is red, and $h'\leq h,k'\leq k$, then $(h',k')$ is also colored red. A Type 1 subset of $S$ has $n$ blue elements with different first member, and a Type 2 subset of $S$ has $n$ red elements with different second member. Show that there are the same number of Type 1 and Type 2 subsets.
Is there any way in which the elements are all not just one color? For example, let $(a,b)$ be colored red and $(c,d)$ be colored blue. Clearly, $0\leq a,c$ and $0\leq b,d$. Hence, $(0,0)$ should be colored both red and blue, which is impossible.
If my explanation is true, then both Type 1 and Type 2 sets cannot exist at the same time, making the whole question confusing
