Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 1 Due January 16, 2014
Assignments are due in class at the beginning of the lecture on the due date. Penalties will
be assessed for late submission, and will increase for repeated late submission. Electronic
copies will b
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 4 Due Wednesday February 26
Assignments are due in class at the beginning of the lecture on the due date. Penalties will
be assessed for late submission, and will increase for repeated late submission. Electronic
copies w
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
INEQUALITIES PROBLEM SET 1
1. Show that if x is any real number,
2
x2 + 1
2
2.
3
x +x+1
2. Prove that for all positive integers n > 1,
1
1
1
1
3
<
+
+ +
< .
2
n+1 n+2
2n
4
3. Prove that for all positive integers n > 1,
1
1
1
1
+
+
+ + 2 > 1.
n n+1 n+2
n
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Some Solutions 6.4
6.4.4 Let Mn be the number of messages of length n in this language. We start by checking the
rst few values, using a as the single letter, and * as the blank, for emphasis.
n
messages
Mn
0
0
1
a
1
2
aa
1
3
aaa,a*
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Some Solutions 7.1
7.1.11 Note that (n4 + 3n2 + 1) n(n3 + 2n) = n2 + 1, then note that (n3 + 2n) n(n2 +
1) = n, and nally that (n2 + 1) n(n) = 1. It follows that any common divisor of
n4 + 3n2 + 1 and n3 + 2n is also a divisor of n2
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Some Solutions 7.3
7.3.14 Well begin with a general observation: except for (2) = 1, the value of (n) is always
even. Proof: if n is a power of 2, i.e. n = 2k with k > 1, then (n) = 2k1 which is
even. If n is greater than 2, but not
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Some Solutions 6.3
6.3.8 If the children are indistinguishable, then the problem is equivalent to nding the number
of solutions in nonnegative integers of x1 + x2 + + x31 = 10, where xi is the number of
10 + 31 1
40
40
cones of avo
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Some Solutions 7.4
7.4.6 (a) If 54 + 55 + 5n = m2 , then 5n = m2 54 (5 1) = m2 2500 = (m 50)(m + 50). Each of these
factors must then be a power of 5, so we require two powers of ve which dier by 100. The powers of 5
are 5, 25, 125,
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Some Solutions 7.2
7.2.6 We are given that a, b and c are integers with a2 + b2 = c2 ; we must show that at least
one of a or b is a multiple of 3, i.e. is congruent to 0 modulo 3. We start by checking
the possible remainders when x
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2024b Quiz 1
All questions have the same value, 5 marks. Do any FOUR questions. You may do
more, but only your best FOUR answers will be counted.
Answers must be written in booklets. No scrap paper other than booklets may be
used. This questio
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Some Solutions 7.5
7.5.9 We are given that p is a prime number. We are to prove that p  (u + p)k uk for all integers u and all
k 1.
1. Use induction.
Solution: Basis: (k = 1) We have (u + p)1 u1 = u + p u = p; therefore p  (u + p)
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b Quiz 1
All questions have the same value, 5 marks. Do any FOUR questions. You may do
more, but only your best FOUR answers will be counted.
Answers must be written in booklets. No scrap paper other than booklets may be
used. This questio
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
The University of Western Ontario
London
Canada
Department of Mathematics
Mathematics 2124b MIDTERM TEST
Friday, March 7, 2014
7 p.m. 9:30 p.m.
The test has two parts, A and B. You must do both parts. All questions have equal value, 10
marks. The test wil
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b Quiz 2
March 26, 2014 50 minutes
All questions have the same value, 5 marks. Do any FOUR questions. You may do more,
but only your best FOUR answers will be counted. All answers must be supported by
appropriate work and explanation. Answ
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
The Ballot Problem and Catalan Numbers
Problem: Candidates A and B receive a and b votes, respectively. If the votes are counted in random
order, what is the probability that A is never behind in the counting?
Solution: Let Pa,b den
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Some Solutions 3.3
3.3.11 Suppose there are n people at the gathering. Each person may know 0, 1, 2, . . . , n 1 of
the other people, so it appears that we have the same number of pigeons (the people) as
pigeonholes (the number of o
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 2 Due Monday January 27, 2014
Assignments are due in class at the beginning of the lecture on the due date. Penalties will
be assessed for late submission, and will increase for repeated late submission. Electronic
copies
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 6 Due Monday March 24
Assignments are due in class at the beginning of the lecture on the due date. Penalties will
be assessed for late submission, and will increase for repeated late submission. Electronic
copies will be
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 3 Due Wednesday February 5
Assignments are due in class at the beginning of the lecture on the due date. Penalties will
be assessed for late submission, and will increase for repeated late submission. Electronic
copies wi
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 1 Solutions
Assignments are due in class at the beginning of the lecture on the due date. Penalties will
be assessed for late submission, and will increase for repeated late submission. Electronic
copies will be accepted
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 7 Due Monday April 7
Assignments are due in class at the beginning of the lecture on the due date. Penalties will
be assessed for late submission, and will increase for repeated late submission. Electronic
copies will be
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 4 Due Wednesday February 26
Assignments are due in class at the beginning of the lecture on the due date. Penalties will
be assessed for late submission, and will increase for repeated late submission. Electronic
copies w
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 7 Due Monday April 7 Solutions
Assignments are due in class at the beginning of the lecture on the due date. Penalties will
be assessed for late submission, and will increase for repeated late submission. Electronic
copie
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 5 Due Monday March 17
Assignments are due in class at the beginning of the lecture on the due date. Penalties will
be assessed for late submission, and will increase for repeated late submission. Electronic
copies will be
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 3 Due Wednesday February 5 Solutions
Assignments are due in class at the beginning of the lecture on the due date. Penalties will
be assessed for late submission, and will increase for repeated late submission. Electronic
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 2 Due Monday January 27, 2014 Solutions
Assignments are due in class at the beginning of the lecture on the due date. Penalties will be
assessed for late submission, and will increase for repeated late submission. Electro
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 5 Due Monday March 17 Solutions
Assignments are due in class at the beginning of the lecture on the due date. Penalties will
be assessed for late submission, and will increase for repeated late submission. Electronic
copi
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
Mathematics 2124b 2014
Assignment 6 Due Monday March 24
Assignments are due in class at the beginning of the lecture on the due date. Penalties will
be assessed for late submission, and will increase for repeated late submission. Electronic
copies will be
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
INEQUALITIES PROBLEM SET 2
1. If xn + y n = z n , where x, y, and z are positive real numbers and n 1, prove that
xy n 1
< .
z2
4
2. Given that x1 + x2 + x3 = 2, nd the maximum value of x1 x2 + x2 x3 + x3 x1 .
2
n
ai
n
a2
i
3. Prove that for any real numb
Western University (Ontario)  Also known as University of Western Ontario
MATHEMATICS
MATH 1224b

Spring 2014
The University of Western Ontario
London
Canada
Department of Mathematics
Mathematics 2124b MIDTERM TEST
Friday, March 7, 2014
7 p.m. 9:30 p.m.
The test has two parts, A and B. You must do both parts. All questions have equal value, 10
marks. The test wil