# Rawles Exam

Professor Thomas Hood Rawles joined the college administration in 1935 and then moved to the mathematics department in 1955. In January of 1959, he gave the department a financial gift in honor of his son, Thomas Post Rawles, who died at an early age. The fund became known as the Rawles fund and was used for a variety of purposes in the department including prizes for mathematics students.

In the late fifties, senior mathematics majors were required to take the Graduate Records Examination. Starting in 1959, the department gave a prize to the students with first and second highest score on that exam. When the block plan began in 1970, the college no longer required comprehensive examinations or the GRE; it was left up to the departments to decide whether the exams were useful. The mathematics department subsequently phased out the exam requirement, but in its place they offered an annual all-college mathematics examination complete with prizes. This became known as the Rawles Exam.

The first Rawles Exam was given in the spring of 1972. Students had three hours to work on any of several problems requiring more mathematical insight than specific mathematical knowledge. The first exam had eight problems, but soon it was traditional to set six problems in the three hours. It is typical for students from throughout the college, not just the mathematics department, to take the exam.

The following problems appeared on the first Rawles Exam in 1972:

1. If a1, a2,...,a7 are integers and b1, b2,...,b7 are the same integers rearranged show that the product, for i = 1 to 7, of all (ai - bi) is even.

2. In some cases the sum of the reciprocals of a set of n different positive integers is equal to one. If n=3, show that there is only one such set and find it. Find also such a set for n=4, 5, and more generally for any value of n &gt; 3.

3. An automobile starts from rest and ends at rest, traversing a distance of one mile in one minute along a straight road. If a governor prevents the speed of the car from exceeding 90 miles per hour, show that at some time of the traverse the acceleration or deceleration of the car was at least 6.6 ft/sec2.

4. A pack of 52 cards is shuffled by starting with the top card and then placing successive cards alternately above and below the growing discard pile. After repeated shuffles of this type, will the pack first return to its original order when the original top card first returns to its top position?

5. Suppose you have a finite number of points in the plane, and every line connecting two of the points has on it a third point of the set. Show that all the points must be on a single line.

6. Suppose x1, x2, x3, ... is a sequence of numbers such that (xn - xn-2) goes to 0 as n goes to infinity. Prove that the limit of (xn - xn-1)/n as n goes to infinity is zero.

7. Show that any curve in the plane of unit length can be covered by a closed rectangle of area 1/4.

8. This famous sequence is defined as follows: f0=1, f1=1, f2=f0+f1=1, and in general, fn=fn-2+fn-1. Consider the ratio rn=fn/fn-1. Assuming that rn has a limit as n approaches infinity, find this limit. Try to prove that in fact the limit does exist.

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