First Annual Iowa Collegiate Mathematics Competition

April 1, 1995
Problems by Elgin Johnston, Iowa State University

1.   Let a be the integer whose base 10 representation consists of 119 ones:

Prove that a is not prime.

2.   Each cube in a collection of  cubes is painted all red or all blue.  The cubes are then used to construct an  cube.  The resulting cube is completely blue on the outside, but every small cube in the interior is red.  The cube is to be reassembled so that the outside surface is completely red.  For what positive integer values of  n can this be done?

3.   An  matrix A satisfies the equation

where I is the identity matrix and Z is the zero matrix.

4.   Find a formula for .

5.   For x > 0 define

Find the value of

6.   Let be real numbers with

Prove that

7.   Vectors a , , and are chosen from such that

(i) For each vector, each of the coordinates is -1 or 1.
(ii) The vectors are mutually perpendicular.

Prove that this can be done only if n is a multiple of 4.

8.   Let

a.   Prove that the equation  has exactly one positive root.

b.   Let r be the positive root of   Prove that if s as any other real or complex root of
      , then

9.   Let E be an ellipse in the plane and let A be a fixed point inside of E. Suppose that two perpendicular lines through A intersect E in points P, P' and Q, Q' respectively. Prove that

is independent of the choice of lines.

10.   Let N₀ be the set of positive integers whose decimal expressions do not contain the digit 0. (Hence but ).   Does the series

converge or diverge? Justify your answer.