Math 289 – Winter 2007 – Problem Set 5
Due February 14, 2007
1.(10) In a sequel of a successful movie The Puzzle, a brother of the Albanian mathematical genius
Le Chiffre, has many secret codes (positive integer numbers with possible very important leading
zeros) which he uses to access his malicious money accounts. James Bond had kindly asked him
to reveal the codes but The Puzzle surprisingly denied his request. Later on, under some light
pressure, The Puzzle agreed to give James Bond one piece of information - a single postive integer
number containing only zeros and ones. Is there any way James Bond can determine all secret
codes? (Is there any way to encode an unknown number of positive integers with possible leading
zeros in a single binary number represented only by zeros and ones and decode them back?)
2.(10)
•Let aand bbe roots of x2+ 45x−26 = 0. Find the values of
a+b, ab, a2+b2, a2b+ab2,|a2−b2|,and a3+b3.
•Let a,b c be roots of x3+ 4x2−9x+ 3 = 0. Find the values of
a+b+c, abc, ab +ac +bc, a2+b2+c2,and a3+b3+c3.
Note that you do not need to solve the cubic equation!
3.(15)
•Determine aand bso, that the polynomial (x−1)2divides polynomial P(x) = ax4+bx3+ 1.
•Prove that (x−1)2divides P(x) = nxn+1 −(n+ 1)xn+ 1 for every positive integer n.
4.(15) Find all polynomials P(x) with integer coefficients which satisfy all the following conditions:
P(2) = 2007 , P (2007) = −325 , P (−325) = 2 .
(Or in general, P(a) = b,P(b) = c, and P(c) = afor three different integers a, b, c.)
5.(20) Let n∈Nbe odd and let k∈Ndivides (n2+ 2). Show that all the possible remainders of
kafter division by 8 are 1 and 3.
6.(25) We call a permutation (x1,x2, . . . , xn) of the numbers 1,2, . . . , 2nquirky if |xi−xi+1 |=n
for at least one i∈ {1,2, . . . , 2n−1}. Prove that more then one-half of all permutations are quirky
for each positive integer. What is probability that a randomly chosen permutation is quirky as
n→ ∞?