Question 4
Let $n\ge2$ be an integer. In an $n\times2n$ board ($n$ rows and $2n$ columns), mark an arbitrary $n^2$ unit cells. Prove that for every integer $k$ satisfying $1\lt k\le\lfloor n/2\rfloor+1$, one can select $k$ rows such that the resulting $k\times2n$ board has at least
$$\frac{k!(n-2k+2)}{(n-k+1)(n-k+2)\cdots(n-1)}$$
columns all of whose cells are marked. Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; this rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
$$\frac{k!(n-2k+2)}{(n-k+1)(n-k+2)\cdots(n-1)}$$
columns all of whose cells are marked. Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; this rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.