Problem 4

Determine all pairs

(h,s)

of positive integers with the following property: if one draws

h

horizontal lines and another

s

lines satisfying (i) they are not horizontal, (ii) no two are parallel, and (iii) no three of the

h+s

lines are concurrent, then the number of regions formed is

1992

Step 5 of 5: Solve the three factor cases

Previous step Next step

s+1=2,11,22\Longrightarrow (h,s)=(995,1),(176,10),(80,21)

Detailed analysis

For $s+1=2$ , the other factor is $1991$ , so $s=1$ and $2h+1=1991$ , giving $h=995$ . For $s+1=11$ , the other factor is $362$ , so $s=10$ and $2h+10=362$ , giving $h=176$ . For $s+1=22$ , the other factor is $181$ , so $s=21$ and $2h+21=181$ , giving $h=80$ . Hence the complete list is $(995,1),(176,10),(80,21)$ .