Seventh Annual Iowa Collegiate Mathematics Competition

Iowa State University
March 31, 2001
Problems by Jerry Heuer of Concordia College

Find all real solutions of the equation .

Here denotes, as usual, the greatest integer less than or equal to x. 

According to Taylor’s Theorem, if is a twice differentiable fucntion on an interval containing a and b, with , then there is a number c between a and b such that 

 .

Express c as simply as possible in terms of a and b if .

Problem 4.  An irrational number.

Let r and s be positive rational numbers withirrational.  Prove that is irrational. 

Problem 5.  Multiplicative inverses.

Let R be the ring of integers modulo 2001.For example, in R, and .

(a)    Determine whether the element 1334 has a multiplicative inverse in R, and if so, find it.  If not show this. 

(b)    Do the same for the element 1333. 

Problem 6.  A harmonic identity.

For each positive integer n, let .

Prove that for every integer ,. 

Problem 7.  Sum of squares divisible by n.
 

A certain set of n integers has the property that the difference between the product of any n – 1 of them and the remaining one is divisible by n.Prove that the sum of the squares of all n integers is divisible by n. 

Problem 8.  Sum the series.
 

Find the sum of the series , and justify your answer. 

Problem 9.  A multiple of 49.
 

Afer serveral applications of the operation of differentiation and the operation of multiplication by x – 1, performed in unspecified order, the polynomial x⁸ +x⁷ is changed to ax + b, where .
 

Prove that a – b is an integer divisible by 49. 

Find all real numbers p such that all three roots of the cubic equation are positive integers.