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The Sun Life Financial
Canadian Open Mathematics Challenge
November 6/7, 2014
STUDENT INSTRUCTION SHEET
General Instructions
1) Do not open the exam booklet until instructed to do so by your supervising
teacher.
2) The supervisor will give you five minutes before the exam starts to fill in the
identification section on the exam cover sheet. You don’t need to rush. Be sure
Mobile phones and
to fill in all information fields and print legibly.
calculators are NOT
3) Once you have completed the exam and given it to your supervising teacher
permitted.
you may leave the exam room.
4) The contents of the COMC 2014 exam and your answers and solutions must not be publicly discussed
(including online) for at least 24 hours.
Exam Format
You have 2 hours and 30 minutes to complete the COMC. There are three sections to the exam:
PART A:
Four introductory questions worth 4 marks each. Partial marks may be awarded for work shown.
PART B:
Four more challenging questions worth 6 marks each. Partial marks may be awarded for work
shown.
PART C:
Four long-form proof problems worth 10 marks each. Complete work must be shown. Partial marks
may be awarded.
Diagrams are not drawn to scale; they are intended as aids only.
Work and Answers
All solution work and answers are to be presented in this booklet in the boxes provided – do not include
additional sheets. Marks are awarded for completeness and clarity. For sections A and B, it is not necessary to
show your work in order to receive full marks. However, if your answer or solution is incorrect, any work that
you do and present in this booklet will be considered for partial marks. For section C, you must show your
work and provide the correct answer or solution to receive full marks.
It is expected that all calculations and answers will be expressed as exact numbers such as 4π, 2 + √7, etc.,
rather than as 12.566, 4.646, etc. The names of all award winners will be published on the Canadian
Mathematical Society web site https://cms.math.ca/comc.
The 2014 Sun Life Financial
Canadian Open Mathematics Challenge
Please print clearly and complete all information below. Failure to print legibly or provide
complete information may result in your exam being disqualified. This exam is not considered
valid unless it is accompanied by your test supervisor’s signed form.
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Page 2 of 16 SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014
Section A
Part A: Question 1 (4 marks)
1. In triangle ABC, there is a point D on side BC such that BA = AD = DC. Suppose
∠BAD = 80◦ . Determine the size of ∠ACB.
A

 
  
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






B
D
C
2 − a = 0 and 3x4 − 48 = 0 have the same real solutions. What is the value
2. The
equations
x
Your Solution:
of a?
3. A positive integer m has the property that when multiplied by 12, the result is a four-digit
Section
Athe form 20A2 for some digit A. What is the 4 digit number, n?
number n of
Alana,
Beatrix,
games
of tennis
together.
the
1.4. In
triangle
ABC,Celine,
there and
is a Deanna
point D played
on side6 BC
such
that BA
= AD In
= each
DC. game,
Suppose
four of =
them
into two
of ∠ACB.
two and one of the teams won the game. If Alana was
Determine
theteams
size of
∠BAD
80◦ .split
on the winning team for 5 games, A
Beatrix for 2 games, and Celine for 1 game, for how many
games was Deanna on the winning 
team?
 
  

Your final answer:















 passes through thepoints (1, 1), (1, 7),
1. The area of the circle that
(9, 1) can be expressed
and



Section B
Part
A:What
Question
as kπ.
is the2 (4
value
of k?
Bmarks)
D
C
−a=
0 andof n for
− 48
= 0 nhave
the+same
solutions.
What is the value
2.2. The
equations
2 + 6n
Determine
all integer
values
which
24 is real
a perfect
square.
of a?
3. 5 Xs and 4 Os are arranged in the below grid such that each number is covered by either an
Solution:
3. Your
AX positive
m has
the of
property
that when
multiplied
12,Osthe
is a four-digit
or an O.integer
There are
a total
126 diﬀerent
ways that
the Xsby
and
canresult
be placed.
Of these
number
n
of
the
form
20A2
for
some
digit
A.
What
is
the
4
digit
number,
n?
126 ways, how many of them contain a line of 3 Os and no line of 3 Xs?
4. Alana,
Beatrix,
Celine,
and
Deanna played
games line,
of tennis
In each
game,
the
A line of
3 in a row
can be
a horizontal
line, a6vertical
or onetogether.
of the diagonal
lines
1−5−9
four
or 7 of
− 5them
− 3. split into two teams of two and one of the teams won the game. If Alana was
on the winning team for 5 games, Beatrix for 2 games, and Celine for 1 game, for how many
1 was 2Deanna
3 on the winning team?
games
x2
3x4
4
5
6
7
8
9
Section B
1. The area of the circle that passes through the points (1, 1), (1, 7), and (9, 1) can be expressed
as kπ. What is the value of k?
2. Determine all integer values of n for which 1n2 + 6n + 24 is a perfect square.
3. 5 Xs and 4 Os are arranged in the below grid such that each number is covered by either an
X or an O. There are a total of 126 diﬀerent ways that the Xs and Os can be placed. Of these
126 ways, how many of them contain a line of 3 Os and no line of 3 Xs?
Your final answer:
A line of 3 in a row can be a horizontal line, a vertical line, or one of the diagonal lines 1−5−9
or 7 − 5 − 3.
cms.math.ca
1
2
3
© 2014 CANADIAN MATHEMATICAL SOCIETY






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
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
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



SUN LIFE FINANCIAL CANADIAN
OPEN
MATHEMATICS
CHALLENGE
2014
B
D
C
x2
− a = 0 and
2. The equations
of
a?
Part A: Question 3 (4 marks)
3x4
Page 3 of 16
− 48 = 0 have the same real solutions. What is the value
3. A positive integer m has the property that when multiplied by 12, the result is a four-digit
number n of the form 20A2 for some digit A. What is the 4 digit number, n?
Solution:
4. Your
Alana,
Beatrix, Celine, and Deanna played 6 games of tennis together. In each game, the
four of them split into two teams of two and one of the teams won the game. If Alana was
on the winning team for 5 games, Beatrix for 2 games, and Celine for 1 game, for how many
games was Deanna on the winning team?
Section B
A
1. The
In triangle
there
a point
D on the
sidepoints
BC such
BAand
= (9,
AD
Suppose
1.
area of ABC,
the circle
thatispasses
through
(1, 1),that
(1, 7),
1) =
canDC.
be expressed
◦
∠BAD
= 80 is. Determine
as
kπ. What
the value ofthe
k? size of ∠ACB.
A
2. Determine all integer values of n forwhich n2 + 6n + 24 is a perfect square.
 
  
3. 5 Xs and 4 Os are arranged in the
such that each number is covered by either an

 below
 grid
that
 ways
X or an O. There are a total of126 diﬀerent
 the Xs and Os can be placed. Of these


 no line of 3 Xs?
126 ways, how many of them
 contain a lineof 3 Os and



of the diagonal lines 1−5−9
A line of 3 in a row can bea horizontal line, a vertical
line,
or one




or 7 − 5 − 3.
2. The1 equations
2
of a?
B
D
C
Your final answer:
4
3 − a = 0 and 3x − 48 = 0 have the same real solutions. What is the value
x2
4
5 integer
6 m has the property that when multiplied by 12, the result is a four-digit
3. A positive
number
n of the form
20A2 for some digit A. What is the 4 digit number, n?
Part A: Question
4 (4 marks)
7
8
9
4. Alana,
Beatrix,
Celine,
and Deanna played 6 games of tennis together. In each game, the
four of them split into two teams of two and one of the teams won the game. If Alana was
on the winning team for 5 games, Beatrix for 2 games, and Celine for 1 game, for how many
games was Deanna on the winning team? 1
Your Solution:
Section B
1. The area of the circle that passes through the points (1, 1), (1, 7), and (9, 1) can be expressed
as kπ. What is the value of k?
2. Determine all integer values of n for which n2 + 6n + 24 is a perfect square.
3. 5 Xs and 4 Os are arranged in the below grid such that each number is covered by either an
X or an O. There are a total of 126 diﬀerent ways that the Xs and Os can be placed. Of these
126 ways, how many of them contain a line of 3 Os and no line of 3 Xs?
A line of 3 in a row can be a horizontal line, a vertical line, or one of the diagonal lines 1−5−9
or 7 − 5 − 3.
1
2
3
4
5
6
© 2014 CANADIAN MATHEMATICAL SOCIETY
7
8
9
Your final answer:
cms.math.ca
four of them split into two teams of two and one of the teams won the game. If Alana was
on the winning team for 5 games, Beatrix for 2 games, and Celine for 1 game, for how many
games was Deanna on the winning team?
Page 4 of 16 SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014
Section B
Part B: Question 1 (6 marks)
1. The area of the circle that passes through the points (1, 1), (1, 7), and (9, 1) can be expressed
as kπ. What is the value of k?
Your Solution:
2. Determine
all integer values of n for which n2 + 6n + 24 is a perfect square.
3. 5 Xs and 4 Os are arranged in the below grid such that each number is covered by either an
X or an O. There are a total of 126 diﬀerent ways that the Xs and Os can be placed. Of these
126 ways, how many of them contain a line of 3 Os and no line of 3 Xs?
A line of 3 in a row can be a horizontal line, a vertical line, or one of the diagonal lines 1−5−9
or 7 − 5 − 3.
1
2
3
4
5
6
7
8
9
1
Your final answer:
cms.math.ca
© 2014 CANADIAN MATHEMATICAL SOCIETY
games was Deanna on the winning team?
Section
BCANADIAN OPEN MATHEMATICS CHALLENGE 2014
SUN LIFE FINANCIAL
Page 5 of 16
1. The area of the circle that passes through the points (1, 1), (1, 7), and (9, 1) can be expressed
Part
Question
(6 marks)
as
kπ.B:What
is the2 value
of k?
2. Determine all integer values of n for which n2 + 6n + 24 is a perfect square.
3. 5Your
Xs Solution:
and 4 Os are arranged in the below grid such that each number is covered by either an
X or an O. There are a total of 126 diﬀerent ways that the Xs and Os can be placed. Of these
126 ways, how many of them contain a line of 3 Os and no line of 3 Xs?
A line of 3 in a row can be a horizontal line, a vertical line, or one of the diagonal lines 1−5−9
or 7 − 5 − 3.
1
2
3
4
5
6
7
8
9
1
Your final answer:
© 2014 CANADIAN MATHEMATICAL SOCIETY
cms.math.ca
Section B
1. The
of the circle that passes through the points
(1, 1),
(1, 7), and
(9, 1) OPEN
can be
expressed CHALLENGE 2014
Page
6 ofarea
16 SUN LIFE
FINANCIAL
CANADIAN
MATHEMATICS
as kπ. What is the value of k?
2. Part
Determine
all integer
values of n for which n2 + 6n + 24 is a perfect square.
B: Question
3 (6 marks)
3. 5 Xs and 4 Os are arranged in the below grid such that each number is covered by either an
X or an O. There are a total of 126 diﬀerent ways that the Xs and Os can be placed. Of these
126 ways, how many of them contain a line of 3 Os and no line of 3 Xs?
A line of 3 in a row can be a horizontal line, a vertical line, or one of the diagonal lines 1−5−9
or 7 − 5 − 3.
1
2
3
4
5
6
7
8
9
Your Solution:
1
Your final answer:
cms.math.ca
© 2014 CANADIAN MATHEMATICAL SOCIETY
SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014
Page 7 of 16
Part B: Question 4 (6 marks)
4. Let f (x) =
x3
1
. Determine the smallest positive integer n such that
+ 3x2 + 2x
f (1) + f (2) + f (3) + · · · + f (n) >
503
.
2014
Your solution:
Section C
1. A sequence of the form {t1 , t2 , ..., tn } is called geometric if t1 = a, t2 = ar, t3 = ar2 , . . . , tn =
arn−1 . For example, {1, 2, 4, 8, 16} and {1, −3, 9, −27} are both geometric sequences. In all
three questions below, suppose {t1 , t2 , t3 , t4 , t5 } is a geometric sequence.
(a) If t1 = 3 and t2 = 6, determine the value of t5 .
(b) If t2 = 2 and t4 = 8, determine all possible values of t5 .
(c) If t1 = 32 and t5 = 2, determine all possible values of t4 .
2. A local high school math club has 12 students in it. Each week, 6 of the students go on a
ﬁeld trip.
(a) Jeﬀrey, a student in the math club, has been on a trip with each other student in the
math club. Determine the minimum number of trips that Jeﬀrey could have gone on.
(b) If each pair of students have been on at least one ﬁeld trip together, determine the
minimum number of ﬁeld trips that could have happened.
3. The line L given by 5y + (2m − 4)x − 10m = 0 in the xy-plane intersects the rectangle with
vertices O(0, 0), A(0, 6), B(10, 6), C(10, 0) at D on the line segment OA and E on the line
segment BC.
(a) Show that 1 ≤ m ≤ 3.
(b) Show that the area of quadrilateral ADEB is
1
3
the area of rectangle OABC.
(c) Determine, in terms of m, the equation of the line parallel to L that intersects OA at F
and BC at G so that the quadrilaterals ADEB, DEGF , F GCO all have the same area.
4. A polynomial f (x) with real coeﬃcients is said to be a sum of squares if there are polynomials
p1 (x), p2 (x), . . . , pn (x) with real coeﬃcients for which
f (x) = p21 (x) + p22 (x) + · · · + p2n (x)
For example, 2x4 + 6x2 − 4x + 5 is a sum of squares because
√
2x4 + 6x2 − 4x + 5 = (x2 )2 + (x2 + 1)2 + (2x − 1)2 + ( 3)2
(a) Determine all values of a for which f (x) = x2 + 4x + a is a sum of squares.
2
© 2014 CANADIAN MATHEMATICAL SOCIETY
Your final answer:
cms.math.ca
4. Let f (x) =
x3 + 3x2 + 2x
. Determine the smallest positive integer n such that
f (1) + f (2) + f (3) + · · · + f (n) >
Page 8 of 16 503
.
2014
SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014
Section C
Section
C 1 (10 marks)
Part C: Question
1. A sequence of the form {t1 , t2 , ..., tn } is called geometric if t1 = a, t2 = ar, t3 = ar2 , . . . , tn =
1. ar
A n−1
sequence
the form{1,
{t2,
...,16}
tn }and
is called
geometric
t1 =
a, tgeometric
= ar2 , . . . ,Intnall
=
1 , t4,
2 ,8,
2 = ar, t3 sequences.
. For of
example,
{1, −3,
9, −27} ifare
both
n−1
.questions
For example,
2, 4, 8, 16}
{1, −3, 9, −27} are both geometric sequences. In all
ar
three
below,{1,
suppose
{t1 , tand
2 , t3 , t4 , t5 } is a geometric sequence.
three questions below, suppose {t1 , t2 , t3 , t4 , t5 } is a geometric sequence of real numbers.
(a) If t1 = 3 and t2 = 6, determine the value of t5 .
(a) If t = 3 and t = 6, determine the value of t5 .
of t5 .
(b) If t21 = 2 and t42 = 8, determine all possible values
Solution: t1 = 3 = a (1 mark) and t2 = ar = 6 , so r = 6/3 = 2. 1 mark.
(c) If t1 = 32 and t5 = 2,4 determine all possible values of t4 .
This gives t5 = 3 × 2 = 48. 1 mark
Your
If solution:
thigh
and t4math
= 8, determine
allstudents
possible in
values
of t5 .week, 6 of the students go on a
2. A(b)
local
club has 12
it. Each
2 = 2school
3
ﬁeld trip.
Solution: t2 = 2 = ar and t4 = 8 = ar , Dividing the two equations gives r2 = 4, so
r = ±2. 1 mark
(a) Jeﬀrey,
in the
has
been on a trip with each other student in the
4
When r a=student
2 we have
a = math
1, so tclub,
5 = 2 = 16. 1 mark.
math club. Determine the minimum number of4 trips that Jeﬀrey could have gone on.
When r = −2 we have a = −1, so t5 = −1 × 2 = −16. 1 mark.
(b) If each pair of students have been on at least one ﬁeld trip together, determine the
(c) If t1 = 32 and t5 = 2, determine all possible values of t4 .
minimum number of ﬁeld trips that could have happened.
1
.1
Solution: We have t1 = 32 = a and t5 = 2 = ar4 . This gives a = 32, and r4 = 16
mark
3. The line L given by 5y + (2m − 4)x − 10m = 0 in the xy-plane intersects the rectangle with
1
1
vertices
O(0,r40),
C(10,
= A(0,
getB(10,
r2 =6),
r2 =0)−1
. D on the line segment OA and E on the line
When
16 we6),
4 or
4 at
−1
2
segment
BC.
When r = 4 , r is not a real number, so this is not a valid sequence. 1 mark
When r2 = 1 we get r = ± 12 . 1 mark.
(a) Show that 14≤ m ≤ 3.1
This gives t4 = 32 × 8 = 4 and t4 = 32 × −1
8 = −4. 1 mark
(b) Show that the area of quadrilateral ADEB is 13 the area of rectangle OABC.
2. The
L given in
byterms
5y + of
(2m
4)xequation
− 10m =of0 the
in the
intersects
the rectangle
with
(c) line
Determine,
m,−the
linexy-plane
parallel to
L that intersects
OA at
F
vertices
O(0,
0),
A(0,
6),
B(10,
6),
C(10,
0)
at
D
on
the
line
segment
OA
and
E
on
the
line
and BC at G so that the quadrilaterals ADEB, DEGF , F GCO all have the same area.
segment BC.
4. A polynomial f (x) with real coeﬃcients is said to be a sum of squares if there are polynomials
m with
≤ 3. real coeﬃcients for which
p2 (x),that
. . . ,1pn≤(x)
p(a)
1 (x),Show
Solution: Since D is on OA, the2x-coordinate
of D is 0.
The y-coordinate D is then the
2
2
(x)
+
p
(x)
+
·
·
·
+
p
(x)
f
(x)
=
p
1
2
n
solution to the equation 5y − 10m = 0, i.e. y = 2m. Hence L intersects OA at D(0, 2m).
For D to be on OA, 0 ≤ 2m ≤ 6, or equivalently 0 ≤ m ≤ 3. 1 mark
For example, 2x4 + 6x2 − 4x + 5 is a sum of squares because
Similarly, the x-coordinate of E is 10, so the y-coordinate is the solution
to 5y + (2m −
√
4 0, whose
2 2m. 2Hence 0 ≤ 82 − 2m ≤2 6 or equivalently
4)(10) − 10m
8−
+ 6x2 − 4xsolutions
+ 5 = (xis2 )y2 =
+ (x
+ 1) + (2x − 1) + ( 3)
2x=
1 ≤ m ≤ 4. 1 mark
2
(a) Determine
am
for≤which
+ a is a sum of squares.
Thus 0 ≤ mall≤values
3 and of
1≤
4 so 1f (x)
≤ m=≤x 3.+14xmark
2
5
cms.math.ca
© 2014 CANADIAN MATHEMATICAL SOCIETY
SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014
© 2014 CANADIAN MATHEMATICAL SOCIETY
Page 9 of 16
cms.math.ca
If high
t1 = 32
and math
t5 = 2,club
determine
possible
of tweek,
4.
2. A (c)
local
school
has 12 all
students
in values
it. Each
6 of the students go on a
1
4
.1
Solution: We have t1 = 32 = a and t5 = 2 = ar . This gives a = 32, and r4 = 16
ﬁeld trip.
mark
1
−1 been on a trip with each other student in the
(a) Jeﬀrey,
the
club,
= 16
weinget
r2 math
= 14 or
r2 =has
When ra4 student
4 .
math
club.
Determine
the
minimum
number
of istrips
that
Jeﬀrey
could 1
have
gone
on.
−1
2
Page 10 of
16 SUN
LIFE
CANADIAN
OPEN
MATHEMATICS
CHALLENGE 2014
notFINANCIAL
a valid
sequence.
mark
When
r = 4 , r is not a real number, so this
1 students have
1 been on at least one ﬁeld trip together, determine the
2 = of
(b) IfWhen
each rpair
4 we get r = ± 2 . 1 mark.
minimum
number
trips that could have
happened.
1
−1
32of
×ﬁeld
This
gives t42=(10
8 = 4 and t4 = 32 × 8 = −4. 1 mark
Part C:
Question
marks)
3.2. The
givenby
by5y
5y++(2m
(2m−−4)x
4)x−−10m
10m==00ininthe
thexy-plane
xy-planeintersects
intersectsthe
therectangle
rectanglewith
with
Theline
lineLLgiven
vertices
O(0,
0),
A(0,
6),
B(10,
6),
C(10,
0)
at
D
on
the
line
segment
OA
and
E
vertices O(0, 0), A(0, 6), B(10, 6), C(10, 0) at D on the line segment OA and Eon
onthe
theline
line
segment
BC.
segment BC.
(a) Show that 1 ≤ m ≤ 3.
(a) Show that 1 ≤ m ≤ 3.
rectangle
OABC.
(b) Show
that the
areaDofis quadrilateral
ADEB is 13 ofthe
Solution:
Since
on OA, the x-coordinate
D area
is 0. of
The
y-coordinate
D is then the
solution to in
theterms
equation
− 10m
= 0, of
i.e.the
y=
2m.
HencetoL Lintersects
OA at OA
D(0,at
2m).
(c) Determine,
of m,5ythe
equation
line
parallel
that intersects
F
ForBC
D to
OA, the
0 ≤quadrilaterals
2m ≤ 6, or equivalently
0 ≤ m, F
≤GCO
3. 1 mark
and
atbe
G on
so that
ADEB, DEGF
all have the same area.
Similarly, the x-coordinate of E is 10, so the y-coordinate is the solution to 5y + (2m −
4. A Your
polynomial
(x) with
coeﬃcients
be2m.
a sum
of squares
there
solution:
4)(10)
−f10m
= 0, real
whose
solutionsisissaid
y =to
8−
Hence
0 ≤ 8 −if 2m
≤ are
6 orpolynomials
equivalently
(x),
.
.
.
,
p
(x)
with
real
coeﬃcients
for
which
p1 (x), 1p2≤
m ≤ 4. n1 mark
2 ≤ 3. 1 mark
Thus 0 ≤ m ≤ 3 and 1 f≤(x)
m=
≤ p42 so
(x)1+≤pm
(x) + · · · + p2 (x)
1
2
n
For example, 2x4 + 6x2 − 4x + 5 is a sum of squares because
√
2x4 + 6x2 − 4x + 5 = (x2 )2 + (x2 + 1)2 + (2x − 1)2 + ( 3)2
(a) Determine all values of a for which f (x) = x2 + 4x + a is a sum of squares.
2
5
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SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014
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Page 11 of 16
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Forline.
example,
{1, 2, 4, 8, 16} and {1, −3, 9, −27} are both geometric sequences. In all
arn−1 .this
1 mark
three questions below, suppose {t1 , t2 , t3 , t4 , t5 } is a geometric 4sequence.
− 2m
The slope of this line is the same as L, so it is given by
. 1 mark
5
(a) IfThus
t1 = the
3 and
linet2is= 6, determine thevalue of t5 .
4 − 2m
Page
12 If
of t16
SUN
FINANCIAL
CANADIAN OPEN MATHEMATICS CHALLENGE 2014
2 and t4 = 8, determiney all
of −
t52m).
.
(b)
= possible values
x +LIFE
(2
2 =
5
(c) If t1 = 32 and t5 = 2, determine all possible values of t4 .
1 mark
Part C: Question 3 (10 marks)
2. A local high school math club has 12 students in it. Each week, 6 of the students go on a
3. ﬁeld
A local
trip.high school math club has 12 students in it. Each week, 6 of the students go on a
ﬁeld trip.
(a) Jeﬀrey, a student in the math club, has been on a trip with each other student in the
(a) math
Jeﬀrey,
a student
in the
club, number
has been
a trip
each
other
student
club.
Determine
themath
minimum
of on
trips
thatwith
Jeﬀrey
could
have
gone in
on.the
math club. Determine the minimum number of trips that Jeﬀrey could have gone on.
(b) If each pair of students have been on at least one ﬁeld trip together, determine the
Solution: There are 11 students in the club other than Jeﬀrey and each ﬁeld trip that
minimum number of ﬁeld trips that could have happened.
Jeﬀrey is on has 5 other students. In order for Jeﬀrey to go on a ﬁeld trip with each
Yourother
solution:
= the
2.2
= 3 ﬁeld
trips 2 marks.
student,
he +
must
 11
50in
3. The line
L given
by 5y
(2mgo
− on
4)xat
− least
10m =
xy-plane
intersects
the rectangle with
vertices O(0, 0), A(0, 6), B(10, 6), C(10, 0) at D on the line segment OA and E on the line
segment BC.
6
(a) Show that 1 ≤ m ≤ 3.
(b) Show that the area of quadrilateral ADEB is
1
3
the area of rectangle OABC.
(c) Determine, in terms of m, the equation of the line parallel to L that intersects OA at F
and BC at G so that the quadrilaterals ADEB, DEGF , F GCO all have the same area.
4. A polynomial f (x) with real coeﬃcients is said to be a sum of squares if there are polynomials
p1 (x), p2 (x), . . . , pn (x) with real coeﬃcients for which
f (x) = p21 (x) + p22 (x) + · · · + p2n (x)
For example, 2x4 + 6x2 − 4x + 5 is a sum of squares because
√
2x4 + 6x2 − 4x + 5 = (x2 )2 + (x2 + 1)2 + (2x − 1)2 + ( 3)2
(a) Determine all values of a for which f (x) = x2 + 4x + a is a sum of squares.
2
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Page 13 of 16
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vertices O(0, 0), A(0, 6), B(10, 6), C(10, 0) at D on the line segment OA and E on the line
segment BC.
(a) Show that 1 ≤ m ≤ 3.
theFINANCIAL
area of rectangle
that the area of quadrilateral ADEBSUN
is 13LIFE
Page (b)
14 ofShow
16 CANADIANOABC.
OPEN MATHEMATICS CHALLENGE 2014
(c) Determine, in terms of m, the equation of the line parallel to L that intersects OA at F
and BC at G so that the quadrilaterals ADEB, DEGF , F GCO all have the same area.
Part C: Question 4 (10 marks)
4. A polynomial f (x) with real coeﬃcients is said to be a sum of squares if there are polynomials
p1 (x), p2 (x), . . . , pn (x) with real coeﬃcients for which
f (x) = p21 (x) + p22 (x) + · · · + p2n (x)
For example, 2x4 + 6x2 − 4x + 5 is a sum of squares because
√
2x4 + 6x2 − 4x + 5 = (x2 )2 + (x2 + 1)2 + (2x − 1)2 + ( 3)2
(a) Determine all values of a for which f (x) = x2 + 4x + a is a sum of squares.
(b) Determine all values of a for which f (x) = x4 + 2x3 + (a − 7)x2 + (4 − 2a)x + a is a sum
2
of squares, and for such values of a, write
f (x) as a sum of squares.
(c) Suppose f (x) is a sum of squares. Prove there are polynomials u(x), v(x) with real
coeﬃcients such that f (x) = u2 (x) + v 2 (x).
Your solution:
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© 2014 CANADIAN MATHEMATICAL SOCIETY
SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014
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Page 16 of 16 SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014
DÉFIOUVERT CANADIEN DE MATHÉMATIQUES FINANCIÈRE SUN LIFE 2014
DÉFIOUVERT
DÉFI OUVERTCANADIEN
CANADIENDE
DEMATHÉMATIQUES
MATHÉMATIQUESFINANCIÈRE
FINANCIÈRESUN
SUNLIFE
LIFE2014
2013
Canadian Mathematical Society
Canadian
Mathematical du
Society
Société mathématique
Canada
Canadian
Canadian Mathematical
Mathematical Society
Society
Société mathématique du Canada
Société
Société mathématique
mathématique du
du Canada
Canada
Page 12 of 16
Page
Page 12
16 of
de16
16
Canadian Open Mathematics Challenge
Défi ouvert canadien de mathématiques
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© 2014 SOCIÉTÉ MATHÉMATIQUE DU CANADA