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 3 of 5: Turn the region condition into a factorization

a_{h,s}=1992\Longleftrightarrow (s+1)(s+2h)=2\cdot1991=2\cdot11\cdot181

Detailed analysis

Substitute $a_{h,s}=1992$ into the region formula and multiply by $2$ . Rearrangement gives $(s+1)(s+2h)=3982=2\cdot11\cdot181$ .