4

Part of 2009 Harvard-MIT Mathematics Tournament

Problems(3)

2009 Algebra #4 - LCM and GCD of Polynomials

Source:

a

b

c

are integers such that the greatest common divisor of

x^2+ax+b

x^2+bx+c

x+1

(in the set of polynomials in

x

with integer coefficients), and the least common multiple of

x^2+ax+b

x^2+bx+c

x^3-4x^2+x+6

a+b+c

number theoryleast common multiplegreatest common divisoralgebrapolynomialquadratics

2009 Calculus #4: Fourth Degree Polynomial

Source:

P

be a fourth degree polynomial, with derivative

P^\prime

P(1)=P(3)=P(5)=P^\prime (7)=0

. Find the real number

x\neq 1,3,5

P(x)=0

calculusalgebrapolynomialderivative

2009 Combinatorics #4 - Number of Functions

Source:

How many functions

f : f\{1, 2, 3, 4, 5\}\longrightarrow\{1, 2, 3, 4, 5\}

f(f(x)) = f(x)

x\in\{ 1,2, 3, 4, 5\}