Fermat II (Probe II) 1999
1) A man was born in the first half of the nineteenth century. He became x years
old on his birthday in the year x2. In what year was he born?
2) A nine digit number is formed by repeating a three digit number three times, for
example, 256,256,256. Show that any such number is always divisible by 1001001.
3) Express the decimal number 33.625 as a sum of integer powers of the number 2, with
no integer power used more than once. (For example, 2.5 = 21 + 2-1.)
4) Professor Proctor administers a test to a class of students, each of whom has his or her
own seat. He tells Professor Adleman the following facts. The room in which the test is
given has 155 seats. In this class, exactly 4/5 of the students are right-handed, exactly
3/4 of the students are males, and exactly 2/3 of the students are using calculators. He
also tells her the number of students in the class. Soon after hearing these facts, Professor
Adleman states that at least 26 students in the class are right-handed males using calculators.
Exactly how many students must be in the class in order for Professor Adleman to reach her
5) Fifteen points lie in a plane in such a way that exactly five of the points are on one straight
line and apart from these five points, no three points lie on a straight line. Find the total
number of distinct straight lines that can be drawn through pairs of the fifteen points.
6) Tom spent all of his money while visiting 5 stores. In each store, he spent $1 more than half
of what he had when he entered the store. How much did Tom have at the outset?
7) The square below is made up of 16 tiles: 4 squares, 4 triangles, and 8 rhombi. All of the squares
and rhombi have sides that are 1 unit long, and the legs of the triangles are also one unit long.
Prove that it is impossible for all of these tiles to be arranged to form an isosceles right triangle.
8) For any integer n > 2, prove that n5-n is divisible by 30.