Math 289 โ Fall 2006 โ Problem Set 12
Due November 29, 2006
1.(10) The Albanian mathematical genius Le Chiffre has three secret codes (positive integer num-
bers) which he uses to access his malicious money accounts. James Bond kindly asked him to reveal
the codes but Le Chiffre surprisingly denied his request. Later on, under some light pressure, Le
Chiffre agreed to give James Bond one piece of information - a single postive integer number. Is
there any way James Bond can determine all three secret codes this way? (Is there any way to
encode three positive integers in a single positive integer and decode them back?)
2.(15) Prove that if g: [0,1] โ[0,1] is a continuous function, then the sequence defined as
xn+1 =g(xn) converges if and only if
lim
nโโ
(xn+1 โxn) = 0 .
3.(25) Two square matrices Aand Bwith real entries satisfy the conditions
A100 =B101 =Iand AB =BA .
Prove that A+B+I, where Idenotes the identity matrix, is invertible.
4.(25) Prove that for any prime number pโฅ5, the number
X
0<k< 2p
3
๎p
k๎
is divisible by p2.
5.(25) Prove that every function of the form
f(x) = a0
2+ cos x+
N
X
n=2
ancos(nx)
with |a0|<1, has positive as well as negative values in the period [0,2ฯ). Also, prove that the
function
F(x) =
100
X
n=1
cos ๎n3
2x๎
has at least 40 zeros in the interval (0,1000).