MAΘ Problem Set 6
October 9, 2003
1. For how many natural numbers a is it true that there are exactly two
perfect squares between a and a + 100 inclusive?
2. In terms of n, what is the ratio of the number of ways to seat n people
in a row to the number of ways to seat them around a circular table?
(Arrangements are considered distinct if they cannot be rotated and made
the same).
3. The area of rectangle ABCD is 1. Let the midpoint of AB be E. Let F
be the intersection of EC and BD. Find the area of triangle F CD.
4. Five is subtracted from both the numerator and the denominator of a
proper fraction that is equal to 3/17. The resulting fraction is equal to
1/n for a positive integer n. Find the sum of all possible values for n.
5. Point A is on the line y = 3x + 1, and the coordinates of B and C are
(0, −3) and (1, 0) respectively. Find the area of triangle ABC.
√
6. Let f (x) = x2 + 2101 x + 2200 . For how many of the first 20 natural
numbers is f (x) an integer?