## NIU Department of Mathematical Sciences Undergraduate Mathematics Contest

Every year since 1987, the faculty of the Department of Mathematical Sciences at Northern Illinois University have hosted an Undergraduate Mathematics Contest, usually in early April.

Contestants select from a mix of elementary and advanced problems based on their class standing, so that underclassmen can (and often have) placed highly in the exam. Students need not be affiliated with the department in any way but must be currently enrolled NIU undergraduates.

Cash prizes are awarded to the top three finishers in May.

Sample problems are listed below. If you are interested in participating, contact Prof. Blau (hblau@niu.edu) early in the spring semester. (You must register in advance.)

## Sample Problems

Underclassmen can answer any questions; upperclassmen are limited in the number of PART A questions they may answer.

PART A samples:

1. If x=1 is a root of the polynomial x3 + a x2-13x + b, find b and each of the other two roots in terms of a.
2. If n is a prime integer and n > 3 , prove that n2 - 1 is evenly divisible by 24.

PART B samples:

1. Find the point on the circle (x-2)2 + (y-4)2 =1 which is farthest from (0,0). Prove your answer.
2. Suppose that y=f(x) is a solution to the differential equation
y'2 - k x y' + k y = 0 (k constant)
Prove that the tangent line to the graph of f at any points (a, f(a)) is also a solution to the equation.