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 4 of 5: Restrict the possible factors

Previous step Next step

s+1\mid3982,\qquad 2\le s+1<s+2h

Detailed analysis

The positive divisors of $3982$ are $1,2,11,22,181,362,1991,3982$ . Since $s>0$ , $s+1\ge2$ ; since $h>0$ , $s+1<s+2h$ . Thus $s+1$ can only be $2$ , $11$ , or $22$ among the divisors.