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
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 inc
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 posi
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
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 divi
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 (
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,
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 o
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
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 sc
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) W
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 sc
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.
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
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
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
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 i
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
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 incr
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 repeat
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 f
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 inc
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
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
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
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,
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
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
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
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.