TITLE: Show that there is a star for which there are less stars in its column than in its row.
QUESTION [5 upvotes]: In a table there are n columns and m rows, n > m. Some cells are
marked by stars, and in each column there's at least one star. Show
that there is a star for which there are less stars in its column than in
its row.

REPLY [2 votes]: While completing @gnometorule's argument was easier, I thought the following approach is also worth mentioning:  
Let $S(i, j)$ denote a function which gives $1$ if there is a star in cell from column $i$ and row $j$, and $0$ otherwise.  Then we have,  
$\sum_{i, j} S(i, j) = X$, the total number of stars.  
Also if $C_i$ denotes the number of stars in column $i$ and $R_j$ denotes the number of stars in row $j$, we have
$\sum_{i=1}^n C_i = \sum_{j=1}^m R_j = X$  
Further, it can be noted that $\sum_j S(i,j) = C_i$ and $\sum_i S(i,j) = R_j$
If all starred cells $(i, j)$ are such that $C_i \ge R_j$,  then
summing over all cells we have:
$\sum_{i, j} (C_i - R_j) S(i, j) \ge 0$  
$\implies \sum_{i, j} C_i S(i, j) \ge \sum_{i, j} R_j S(i, j) $  
$\implies \sum_{i} C_i C_i \ge \sum_{j} R_j R_j $  
$\implies \sum_{i=1}^n C_i^2 \ge \sum_{j=1}^m R_j^2 $  
Now, I believe the above inequality is not possible for such column and row sums, i.e. if:
$0 < C_i \le m$ and $0 \le R_j \le n$ with $n > m$, but have not been able to prove it.