Work and Answers All solution work and answers are to be presented in this booklet in the boxes provided – do not include
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 sit
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.First Name: Grade:
8 9 10 11 12 Cégep Last Name: Other: T-Shirt Size: (Optional. For prize draw)
Are you currently registered in full-time attendance at an elementary, secondary or
XS S M Cégep school, or home schooled and have been since September 15th of this year? L XL XXL Yes No
Date of Birth: y y y y m m d d A re you a Canadian Citizen or a Permanent Resident of Canada (regardless of current
Gender: (Optional) address)? Yes No
Male Female E-mail Address:
Signature: ________________________________ For official use only:
A3 A4 B3 B4 C3 C4 TOTAL A1 A2 B1 B2 C1 C2
Marker initials Data entry initials Verification initials Page 2 of 16 SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014 Section A
1. In triangle ABC, there is a point D on side BC such that BA = AD = DC. Suppose ∠BAD = 80
2
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.
B A C D
2. The equations x
− a = 0 and 3x
8
4
− 48 = 0 have the same real solutions. What is the value of a?
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?
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?
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 n
2 + 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
Your Solution: Your Solution: Your final answer: Your final answer: Part A: Question 1 (4 marks)
Part A: Question 2 (4 marks)
9
7
◦
4
. Determine the size of ∠ACB.
B A C D
2. The equations x
2
− a = 0 and 3x
− 48 = 0 have the same real solutions. What is the value of a?
6
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?
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?
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 n
2 + 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
1 Section A
SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014 Page 3 of 16
− 48 = 0 have the same real solutions. What is the value of a?
. Determine the size of ∠ACB.
B A C D
2. The equations x
2
− a = 0 and 3x
4
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?
1. In triangle ABC, there is a point D on side BC such that BA = AD = DC. Suppose ∠BAD = 80
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?
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 n
2 + 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
Your Solution: Your final answer: Part A: Question 3 (4 marks) Part A: Question 4 (4 marks) Your final answer:
◦
9
− 48 = 0 have the same real solutions. What is the value of a?
B C D
2. The equations x
2
− a = 0 and 3x
4
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?
8
2
7
6
5
4
3
1
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?
−5−9 or 7 − 5 − 3.
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
2 + 6n + 24 is a perfect square.
2. Determine all integer values of n for which n
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?
Section B
1 Section A
6 Your Solution:
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?
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?
2 Your Solution: 2. Determine all integer values of n for which n + 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:
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
SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014 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 B: Question 2 (6 marks)
as kπ. What is the value of k?
2 2. Determine all integer values of n for which n + 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
Your Solution:
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:
Section B
1. The area of the circle that passes through the points (1, 1), (1, 7), and (9, 1) can be expressed
Page 6 of 16 SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014
as kπ. What is the value of k?
2 2. Determine all integer values of n for which n + 6n + 24 is a perfect square.
Part 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:
SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014 Page 7 of 16
Part B: Question 4 (6 marks)1
4. Let f (x) = . Determine the smallest positive integer n such that
3
2
x + 3x + 2x 503 f (1) + f (2) + f (3) + . · · · + f(n) >
2014
Your solution: Section C
2
1. A sequence of the form , t , ..., t = a, t = ar, t = ar , . . . , t =
1 2 n
1
2 3 n
{t } is called geometric if t
n −1
ar . For example, {1, 2, 4, 8, 16} and {1, −3, 9, −27} are both geometric sequences. In all three questions below, suppose , t , t , t , t
{t } is a geometric sequence. (a) If t = 3 and t = 6, determine the value of t .
1
2
5 (b) If t = 2 and t = 8, determine all possible values of t .
2
4
5 (c) If t = 32 and t = 2, determine all possible values of t .
1
5
4
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.
1
(b) Show that the area of quadrilateral ADEB is the area of rectangle OABC.
3
(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 p
1 (x), p 2 (x), . . . , p n (x) with real coefficients for which
2
2
2
f (x) = p (x) + p (x) + (x)
1 2 · · · + p n
4
2 For example, 2x + 6x
− 4x + 5 is a sum of squares because √
4
2
2
2
2
2
2
2
2x + 6x ) + (x + 1) + (2x + ( 3) − 4x + 5 = (x − 1)
2 (a) Determine all values of a for which f (x) = x + 4x + a is a sum of squares.
2 Your final answer:
4. Let f (x) = . Determine the smallest positive integer n such that
3
2
x + 3x + 2x 503 f (1) + f (2) + f (3) + .
· · · + f(n) > 2014
Page 8 of 16 SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014 Section C Section C
Part C: Question 1 (10 marks)
2
1. A sequence of the form , t , ..., t = a, t = ar, t = ar , . . . , t =
1 2 n
1
2 3 n
{t } is called geometric if t
2 n −1
1. A sequence of the form
1 , t 2 , ..., t n 1 = a, t 2 = ar, t 3 = ar , . . . , t n =
{t } is called geometric if t ar . For example, {1, 2, 4, 8, 16} and {1, −3, 9, −27} are both geometric sequences. In all
n −1
ar . For example, {1, 2, 4, 8, 16} and {1, −3, 9, −27} are both geometric sequences. In all three questions below, suppose , t , t , t , t
1
2
3
4
5 {t } is a geometric sequence.
three questions below, suppose
1 , t 2 , t 3 , t 4 , t
5 {t } is a geometric sequence of real numbers.
(a) If t = 3 and t = 6, determine the value of t .
1
2
5 (a) If t = 3 and t = 6, determine the value of t .
1
2
5 (b) If t = 2 and t = 8, determine all possible values of t .
2
4
5 Solution: t = 3 = a (1 mark) and t = ar = 6 , so r = 6/3 = 2. 1 mark.
1
2 (c) If t = 32 and t = 2, determine all possible values of t .
1
5
4
4 This gives t = 3 = 48. 1 mark
5
× 2
Your solution: 2. A local high school math club has 12 students in it. Each week, 6 of the students go on a (b) If t = 2 and t = 8, determine all possible values of t .
2
4
5
3
2 field trip.
Solution: t = 2 = ar and t = 8 = ar , Dividing the two equations gives r = 4, so
2
4
r = (a) Jeffrey, a student in the math club, has been on a trip with each other student in the
4 When r = 2 we have a = 1, so t = 2 = 16. 1 mark.
5 math club. Determine the minimum number of trips that Jeffrey could have gone on.
4
When r = = =5 −2 we have a = −1, so t −1 × 2 −16. 1 mark.
(b) If each pair of students have been on at least one field trip together, determine the (c) If t = 32 and t = 2, determine all possible values of t .
1
5
4 minimum number of field trips that could have happened.
4
4
1 Solution: We have t = 32 = a and t = 2 = ar . This gives a = 32, and r = . 1
1
5
16
mark
3. The line L given by 5y + (2m − 4)x − 10m = 0 in the xy-plane intersects the rectangle with
4
1
2
1
2 −1
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 When r = we get r = or r = .
16
4
4 2 −1
segment BC.
When r = , r is not a real number, so this is not a valid sequence. 1 mark
4
2
1
1 When r = we get r = . 1 mark.
±
4
2
(a) Show that 1 ≤ m ≤ 3.
1 −1
This gives t = 32 = 4 and t = 32 =
4 ×
4 × −4. 1 mark
8
8
1
(b) Show that the area of quadrilateral ADEB is the area of rectangle OABC.
3
2. The line L given by 5y + (2m (c) Determine, in terms of m, the equation of the line parallel to L that intersects OA at F
− 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 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 (a) Show that 1 p (x), p (x), . . . , p (x) with real coefficients for which
1 2 n ≤ m ≤ 3.
Solution: Since D is on OA, the x-coordinate of D is 0. The y-coordinate D is then the
2
2
2
f (x) = p (x) + p (x) + (x)
1
2 · · · + p 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
4 2 ≤ 2m ≤ 6, or equivalently 0 ≤ m ≤ 3. 1 mark
For example, 2x + 6x − 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
2
2
2
2
2
2
2
4)(10) 2x + 6x ) + (x + 1) + (2x + ( 3) − 10m = 0, whose solutions is y = 8 − 2m. Hence 0 ≤ 8 − 2m ≤ 6 or equivalently
− 4x + 5 = (x − 1)
1 ≤ m ≤ 4. 1 mark
2 (a) Determine all values of a for which f (x) = x + 4x + a is a sum of squares.
Thus 0 ≤ m ≤ 3 and 1 ≤ m ≤ 4 so 1 ≤ m ≤ 3. 1 mark
2
5 SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014 Page 9 of 16
(c) If t
1 = 32 and t 5 = 2, determine all possible values of t
4 .
2. A local high school math club has 12 students in it. Each week, 6 of the students go on a
4
4
1
Solution: We have t = 32 = a and t = 2 = ar . This gives a = 32, and r = . 1 field trip.
1
5
16
mark
4
1
2
1
2 −1
(a) Jeffrey, a student in the math club, has been on a trip with each other student in the When r = we get r = or r = .
16
4
4 math club. Determine the minimum number of trips that Jeffrey could have gone on. 2 −1 Page 10 of 16 SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014
When r = , r is not a real number, so this is not a valid sequence. 1 mark
4
1
1
2
(b) If each pair of students have been on at least one field trip together, determine the When r = we get r = . 1 mark.
±
4
2 minimum number of field trips that could have happened.
1 −1
This gives t = 32 = 4 and t = 32 =
4
4
× × −4. 1 mark
8
8 Part C: Question 2 (10 marks)
3. The line L given by 5y + (2m − 4)x − 10m = 0 in the xy-plane intersects the rectangle with
2. 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 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. segment BC.
(a) Show that 1 ≤ m ≤ 3.
(a) Show that 1 ≤ m ≤ 3.
1
(b) Show that the area of quadrilateral ADEB is the area of rectangle OABC.3 Solution: Since D is on OA, the x-coordinate of D is 0. The y-coordinate D is then the
(c) Determine, in terms of m, the equation of the line parallel to L that intersects OA at F solution to the equation 5y − 10m = 0, i.e. y = 2m. Hence L intersects OA at D(0, 2m). and BC at G so that the quadrilaterals ADEB, DEGF , F GCO all have the same area. For D to be on OA, 0
≤ 2m ≤ 6, or equivalently 0 ≤ m ≤ 3. 1 mark Similarly, the x-coordinate of E is 10, so the y-coordinate is the solution to 5y + (2m
−
4. A polynomial f (x) with real coefficients is said to be a sum of squares if there are polynomials
Your solution:
4)(10) − 10m = 0, whose solutions is y = 8 − 2m. Hence 0 ≤ 8 − 2m ≤ 6 or equivalently
1 2 n
1 ≤ m ≤ 4. 1 mark
2
2
≤ m ≤ 3 and 1 ≤ m ≤ 4 so 1 ≤ m ≤ 3. 1 mark f (x) = p (x) + p (x) + (x)
2 Thus 0
1
2 · · · + p n
4
2 For example, 2x + 6x
− 4x + 5 is a sum of squares because √
4
2
2
2
2
2
2
2
2x + 6x ) + (x + 1) + (2x + ( 3) − 4x + 5 = (x − 1)
2 (a) Determine all values of a for which f (x) = x + 4x + a is a sum of squares.
2
5
SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014 Page 11 of 16
· 10 = 10b, so b = 2. Hence the point M(5, 2) is on
n −1
ar . For example, {1, 2, 4, 8, 16} and {1, −3, 9, −27} are both geometric sequences. In all this line. 1 mark three questions below, suppose , t , t , t , t
4 {t
1
2
3
4 5 } is a geometric sequence.
− 2m The slope of this line is the same as L, so it is given by . 1 mark
5 (a) If t = 3 and t = 6, determine the value of t .
Thus the line is
1
2
5
4 − 2m
Page 12 of 16 SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014 (b) If t = 2 and t = 8, determine all possible values of t y = x + (2 .
2
4
5 − 2m).
5 (c) If t
1 = 32 and t 5 = 2, determine all possible values of t 4 .
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. A local high school math club has 12 students in it. Each week, 6 of the students go on a field trip.
field trip. (a) Jeffrey, a student in the math club, has been on a trip with each other student in the
(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. 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
other student, he must go on at least = 2.2 = 3 field trips 2 marks.
11 Your solution:
5
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.
6 (a) Show that 1 ≤ m ≤ 3.
1
(b) Show that the area of quadrilateral ADEB is the area of rectangle OABC.
3
(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 p (x), p (x), . . . , p (x) with real coefficients for which
1 2 n
2
2
2
f (x) = p (x) + p (x) + (x)
1
2 · · · + p n
4
2 For example, 2x + 6x
− 4x + 5 is a sum of squares because √
4
2
2
2
2
2
2
2
2x + 6x ) + (x + 1) + (2x + ( 3) − 4x + 5 = (x − 1)
2 (a) Determine all values of a for which f (x) = x + 4x + a is a sum of squares.
2
SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014 Page 13 of 16 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.
1
(b) Show that the area of quadrilateral ADEB is the area of rectangle OABC.Page 14 of 16 SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014
3
(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 p (x), p (x), . . . , p (x) with real coefficients for which
1 2 n
2
2
2
f (x) = p (x) + p (x) + (x)
1 2 · · · + p n
4
2 For example, 2x + 6x
− 4x + 5 is a sum of squares because √
4
2
2
2
2
2
2
2
2x + 6x ) + (x + 1) + (2x + ( 3) − 4x + 5 = (x − 1)
2 (a) Determine all values of a for which f (x) = x + 4x + a is a sum of squares.
4
3
2
(b) Determine all values of a for which f (x) = x + 2x + (a + (4 − 7)x − 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
2
2 coefficients such that f (x) = u (x) + v (x).
Your solution:
SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014 Page 15 of 16 Page 16 of 16 SUN LIFE FINANCIAL CANADIAN OPEN MATHEMATICS CHALLENGE 2014 Canadian Open Mathematics Challenge Canadian Mathematical Society Société mathématique du Canada
DÉFI OUVERT CANADIEN DE MATHÉMATIQUES FINANCIÈRE SUN LIFE 2013 Page 16 de 16 Défi ouvert canadien de mathématiques