November 14, 1998
9:00 a.m. to 12:00 noon

To the team members: With any luck you will find these problems fun as well as challenging. You should probably not expect to work all ten of them in the allotted time. Each problem counts 10 points. While partial credit will be given for significant progress or for significant partial solutions, a thorough job on a few of them will be better than some exploratory work on all, so try to pick the ones which are most appealing to you and on which you think you can make some significant progress.


Each team may submit one solution to each problem. Think of your solution as an essay; a logical argument which makes clear why your answer to the question is correct, or why the assertion whose proof is called for in the problem is true.

PLEASE BEGIN EACH PROBLEM ON A NEW SHEET OF PAPER. Team identification and problem number should be clearly given at the top of each sheet of paper submitted.

1. Matching pennies.
Adolph and Bertha amused themselves for a while by matching pennies. On each toss, Adolph won a penny from Bertha if their coins matched, and Bertha won one from Adolph if they failed to match. When they stopped, their coins had matched 13 times and Bertha had gained 8 pennies. How many times did they toss? (Don't forget to explain your answer!)

2. Base b quadratic.
In base b, where b > 9, the quadratic equation x2-mx+n=0  has roots 9 and 5. If the base b representation of m is 11, what is the base b representation of n?

3. Hyperbolics.
Recall that the hyperbolic functions  sinh and  cosh are defined by

Suppose that

sinh  2x  +  cosh  2x  = 2.


sinh 9x + cosh 9x,

and justify your answer.

4. A relatively prime sequence.

5. Sum the series.
Find the sum of the series

and justify your answer.

6. A circle and some points.
A unit circle and 1998 distinct points Pi are chosen in a plane. Prove that there is a point Q on the circle such that the sum of the distances from Q to the 1998 points Pi is greater than 1998.

7. Equal integrals.
Let f(x) be a polynomial of degree 2 and g(x) a polynomial of degree 3 such that f(x)=g(x) at some three distinct equally spaced points a, (a+b)/2 and b. Prove that

8. Reciprocal inequality.
Suppose that a,b,c and d are nonzero real numbers such that  Prove that

9. Order of an element.
In a certain group G with identity e, there are two elements x and y different from e satisfying x5= e and  xyx-1=y2.
What is the order of y; i.e., what is the smallest positive integer n such that yn=e?

10. Not a fifth power.
Suppose N is a number of the form