FLSAM Region Team Round

Directions: You have two hours to complete this test, and no calculators or other aids may be used.The problems are intended to be challenging, even for the best students, so do not be discouraged if youcant solve many of them. In fact, we do not expect test-takers to be able to solve all of the problemswithin the time constraints: this test is also designed to be a learning experience! All we ask is thatyou give each problem your best effort. Be sure to write your answers on the provided answer sheet.Good luck, and have fun!

Note that the questions with point values of 3 and 6 merely need a numerical answer; however, thequestions with point values of 12 and 24 require an explanation/proof that is logically sound.

1 (3 Points Each)

A-1: Find all possible values of f(2013), where f(x) is a polynomial such that xf(x 1) =(x + 1)f(x).

C-1: G-1: NT-1: Find all integer x such that x2 + 2x 1 and x2 + 4x + 6 are both primes.

2 (6 Points Each)

A-2: C-2: G-2: NT-2:

3 (12 Points Each)

A-3:

C-3: In a 3 3 square, each square is shaded/unshaded at random (with probability 12

). What

is the probability that no 2 2 square is fully shaded? G-3: N-3: Consider 3 positive integers a, b, c such that a|c2|b4|a8. If m = lcm(a, b, c) and n =

gcd(a, b, c), evaluate lcm

(a2

b,b2

c,c2

a

)using only a, b, c,m, n.

1

FLSAM Region Team Round

4 (24 Points Each)

A-4: Let C denote the set of all complex numbers with both integral real and imaginary parts,and let a and b be two relatively prime integers. Let two complex numbers x and y in C bedenoted as a + bi equivalent if x y = (a + bi)(c) for some c in C. Additionally, a set A isdenoted as a C-class if it is the set of all numbers in C that are a+ bi equivalent. Prove thatthe number of C-classes is a2 + b2.

C-4: Alex is climbing a set of infinite stairs. He starts at the bottom and first moves up to thefirst step. Afterwards, he moves either up or down one step with equal probability, and will alwayscontinue until he reaches the bottom again. What is the probability that Alex stops?

G-4: In a convex pentagon ABCDE, we have that AB||CE, BC||DA, CD||EB, DE||AC, andEA||BD. Given that AB = 1, prove that CE = 1 +

5

2.

N-4: Let r2(n) be the number of ways that n can be written as the sum of 2 nonnegative squares.What is the expected value of r2(n) over all positive integers?

2