CSMC - Problems (2015 - 2011).pdf

8/18/2019 CSMC - Problems (2015 - 2011).pdf

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The CENTRE for EDUCATION

in MATHEMATICS and COMPUTING

cemc.uwaterloo.ca

Canadian Senior Mathematics Contest

Wednesday, November 25, 2015(in North America and South America)

Thursday, November 26, 2015(outside of North America and South America)

Time: 2 hours c2015 University of WaterlooCalculators are allowed, with the following restriction: you may not use a device

that has internet access, that can communicate with other devices, or that contains

previously stored information. For example, you may not use a smartphone or a

tablet.

Do not open this booklet until instructed to do so.There are two parts to this paper. The questions in each part are arranged roughly in order of increasing difficulty. The early problems in Part B are likely easier than the later problems inPart A.

PART A

1. This part consists of six questions, each worth 5 marks.

2. Enter the answer in the appropriate box in the answer booklet.For these questions, full marks will be given for a correct answer which is placed in the box.Part marks will be awarded only if relevant work is shown in the space provided in theanswer booklet.

PART B

1. This part consists of three questions, each worth 10 marks.

2. Finished solutions must be written in the appropriate location in the answerbooklet. Rough work should be done separately. If you require extra pages for yourfinished solutions, they will be supplied by your supervising teacher. Insert these pages intoyour answer booklet. Be sure to write your name, school name and question number on anyinserted pages.

3. Marks are awarded for completeness, clarity, and style of presentation. A correct solution,poorly presented, will not earn full marks.

At the completion of the contest, insert your student information form inside your

answer booklet.

Do not discuss the problems or solutions from this contest online for the next 48 hours.

The name, grade, school and location, and score range of some top-scoring students will be

published on the Web site, cemc.uwaterloo.ca. In addition, the name, grade, school and location,

and score of some students may be shared with other mathematical organizations for other

recognition opportunities.


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NOTE:1. Please read the instructions on the front cover of this booklet.2. Write solutions in the answer booklet provided.

3. Express calculations and answers as exact numbers such as π + 1 and√

2, etc.,

rather than as 4.14 . . . or 1.41 . . ., except where otherwise indicated.4. While calculators may be used for numerical calculations, other mathematical

steps must be shown and justified in your written solutions and specific marksmay be allocated for these steps. For example, while your calculator might beable to find the x-intercepts of the graph of an equation like y = x3 − x, youshould show the algebraic steps that you used to find these numbers, rather thansimply writing these numbers down.

5. Diagrams are not drawn to scale. They are intended as aids only.6. No student may write both the Canadian Senior Mathematics Contest and the

Canadian Intermediate Mathematics Contest in the same year.

PART A

For each question in Part A, full marks will be given for a correct answer which is placed inthe box. Part marks will be awarded only if relevant work is shown in the space providedin the answer booklet.

1. If 8

24 =

4

x + 3, what is the value of x?

2. Let A, B and C be non-zero digits, so that BC is a two-digit positive integer andABC is a three-digit positive integer made up of the digits A, B and C . Supposethat

B C A B C

+ A B C

8 7 6

What is the value of A + B + C ?

3. A 5 m × 5 m flat square roof receives 6 mm of rainfall. All of this water (and noother water) drains into an empty rain barrel. The rain barrel is in the shape of acylinder with a diameter of 0.5 m and a height of 1 m. Rounded to the nearest tenthof a percent, what percentage of the barrel will be full of water?

4. Determine all values of x for which 2 · 4x2−3x2 = 2x−1.

5. In a psychology experiment, an image of a cat or an image of a dog is flashed brieflyonto a screen and then Anna is asked to guess whether the image showed a cat ora dog. This process is repeated a large number of times with an equal number of images of cats and images of dogs shown. If Anna is correct 95% of the time whenshe guesses “dog” and 90% of the time when she guesses “cat”, determine the ratioof the number of times she guessed “dog” to the number of times she guessed “cat”.


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6. Suppose that X and Y are angles with tan X = 1

m and tan Y =

a

n for some positive

integers a, m and n. Determine the number of positive integers a ≤ 50 for whichthere are exactly 6 pairs of positive integers (m, n) with X + Y = 45◦.

(Note: The formula tan(X + Y ) = tan X + tan Y

1− tan X tan Y may be useful.)

PART B

For each question in Part B, your solution must be well organized and contain words of explanation or justification. Marks are awarded for completeness, clarity, and style of presentation. A correct solution, poorly presented, will not earn full marks.

1. The line y = 2x + 4 intersects the y-axis at R, as shown. A second line, parallel tothe y-axis, is drawn through P ( p, 0), with p > 0. These two lines intersect at Q.

(a) Determine the length of OR.(Note that O is the origin (0, 0).)

(b) Determine the coordinates of point Q interms of p.

(c) If p = 8, determine the area of OPQR.

(d) If the area of OPQR is 77, determinethe value of p.

y

x P ( p, 0)

Q

O

R

2. (a) If f (x) = x

x− 1 for x = 1, determine all real numbers r = 1 for which f (r) = r.(b) If f (x) =

x

x − 1 for x = 1, show that f (f (x)) = x for all real numbers x = 1.

(c) Suppose that k is a real number. Define g(x) = 2x

x + k for x = −k. Determine

all real values of k for which g (g(x)) = x for every real number x with x = −kand g(x) = −k.

(d) Suppose that a, b and c are non-zero real numbers. Define h(x) = ax + b

bx + c for

x = − cb

. Determine all triples (a,b,c) for which h(h(x)) = x for every real

number x with x = −c

b and h(x) = −c

b .


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3. Given a sequence a1, a2, a3, . . . of positive integers, we define a new sequenceb1, b2, b3, . . . by b1 = a1 and, for every positive integer n ≥ 1,

bn+1 =

bn + an+1 if bn ≤ an+1bn − an+1 if bn > an+1

For example, when a1, a2, a3, · · · is the sequence 1, 2, 1, 2, 1, 2, . . . we have

n 1 2 3 4 5 6 7 8 9 10 · · ·an 1 2 1 2 1 2 1 2 1 2 · · ·bn 1 3 2 4 3 1 2 4 3 1 · · ·

(a) Suppose that an = n2 for all n ≥ 1. Determine the value of b10.

(b) Suppose that an = n for all n ≥ 1. Determine all positive integers n withn


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cemc.uwaterloo.ca


Thursday, November 20, 2014(in North America and South America)

Friday, November 21, 2014(outside of North America and South America)

Time: 2 hours c2014 University of WaterlooCalculators are allowed, with the following restriction: you may not use a device

that has internet access, that can communicate with other devices, or that contains

previously stored information. For example, you may not use a smartphone or a

tablet.

Do not open this booklet until instructed to do so.There are two parts to this paper. The questions in each part are arranged roughly in order of increasing difficulty. The early problems in Part B are likely easier than the later problems inPart A.

PART A



PART B





answer booklet.



published on the Web site, cemc.uwaterloo.ca. In addition, the name, grade, school and location,

and score of some students may be shared with other mathematical organizations for other

recognition opportunities.


6/20


NOTE:1. Please read the instructions on the front cover of this booklet.2. Write solutions in the answer booklet provided.3. 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 . . . or 4.646 . . ..4. While calculators may be used for numerical calculations, other mathematical

steps must be shown and justified in your written solutions and specific marksmay be allocated for these steps. For example, while your calculator might beable to find the x-intercepts of the graph of an equation like y = x3 − x, youshould show the algebraic steps that you used to find these numbers, rather thansimply writing these numbers down.

5. Diagrams are not drawn to scale. They are intended as aids only.6. No student may write both the Canadian Senior Mathematics Contest and the

Canadian Intermediate Mathematics Contest in the same year.

PART A


1. In the diagram, ABCD is a square, ABE isequilateral, and AEF is equilateral. What is themeasure of ∠DAF ?

A

B

C

D

E

F

2. In a jar, the ratio of the number of dimes to the number of quarters is 3 : 2. If thetotal value of these coins is $4, how many dimes are in the jar?

(Each dime is worth 10 cents, each quarter is worth 25 cents, and $1 equals 100cents.)

3. Positive integers m and n satisfy mn = 5000. If m is not divisible by 10 and n isnot divisible by 10, what is the value of m + n?

4. A function f satisfies f (x) + f (x + 3) = 2x + 5 for all x.If f (8) + f (2) = 12, determine the value of f (5).

5. Determine all real numbers x for which 3

(2 + x)2

+ 3 3

(2− x)2

= 4 3

√ 4− x2

.

6. Ten lockers are in a row. The lockers are numbered in order with the positive integers1 to 10. Each locker is to be painted either blue, red or green subject to the followingrules:

• Two lockers numbered m and n are painted different colours whenever m − nis odd.

• It is not required that all 3 colours be used.In how many ways can the collection of lockers be painted?


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PART B

For each question in Part B, your solution must be well organized and contain words of explanation or justification. Marks are awarded for completeness, clarity, and style of presentation. A correct solution, poorly presented, will not earn full marks.

1. An arithmetic sequence is a sequence in which each term after the first is obtainedfrom the previous term by adding a constant. For example, 5, 7, 9 is an arithmeticsequence with three terms.

(a) Three of the five numbers 2, 5, 10, 13, 15 can be chosen to form an arithmeticsequence with three terms. What are the three numbers?

(b) The numbers p, 7, q, 13, in that order, form an arithmetic sequence with fourterms. Determine the value of p and the value of q .

(c) The numbers a,b,c, (a + 21), in that order, form an arithmetic sequence withfour terms. Determine the value of c− a.

(d) The numbers (y

−6), (2y+3), (y2+2), in that order, form an arithmetic sequence

with three terms. Determine all possible values of y.

2. (a) (i) In the diagram, semi-circles are drawn on thesides of right-angled XY Z , as shown. If the area of the semi-circle with diameter Y Z is 50π and the area of the semi-circle withdiameter XZ is 288π, determine the area of the semi-circle with diameter XY .

X Y

Z

x y

z

(ii) In the diagram, squares with side

lengths p, q and r are constructed onthe sides of right-angled P QR, asshown. Diagonals QT , RU and P S have lengths a, b and c, respectively.Prove that the triangle formed withsides of lengths a, b and c is a right-angled triangle. P Q

R

S

T U

pq

r

b

a

c

(b) In the diagram, XY Z is right-angled atZ and AZ is the altitude from Z to X Y .Also, segments AD and AB are altitudesin AXZ and AY Z , respectively, andsegments DE and BC are altitudes inADX and ABY , respectively. Provethat AE = AC .

X Y

Z

E A C

B

D


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3. For any real number x, x denotes the largest integer less than or equal to x.For example, 4.2 = 4 and −2.4 = −3. That is, x is the integer that satisfiesthe inequality x ≤ x < x+ 1.

(a) The equation x2 = 3x + 1 has two solutions. One solution is x = √ 7. Thesecond solution is of the form x =

√ a for some positive integer a. Determine

the value of a.

(b) For each positive integer n, determine all possible integer values of the expressionx2 − 3x, where x is a real number with x = n.

(c) For each integer k with k ≥

0, determine all real numbers x for whichx2 = 3x+ (k2 − 1).


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cemc.uwaterloo.ca

CanadianSenior

Mathematics ContestThursday, November 21, 2013

(in North America and South America)

Friday, November 22, 2013 (outside of North America and South America)

Time: 2 hours c2013 University of WaterlooCalculators are permitted, provided they are non-programmable and without graphicdisplays.

Do not open this booklet until instructed to do so.There are two parts to this paper.

PART A


2. Enter the answer in the appropriate box in the answer booklet.For these questions, full marks will be given for a correct answer which is placed in the box.

Part marks will be awarded only if relevant work is shown in the space provided in theanswer booklet.

PART B




NOTES:

The questions in each part are arranged roughly in order of increasing difficulty.

The early problems in Part B are likely easier than the later problems in Part A.


answer booklet.



published on the Web site, http://www.cemc.uwaterloo.ca. In addition, the name, grade, school

and location, and score of some students may be shared with other mathematical organizations

for other recognition opportunities.


10/20


NOTE: 1. Please read the instructions on the front cover of this booklet.2. Write solutions in the answer booklet provided.3. 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 . . .

or 4.646 . . ..4. Calculators are permitted, provided they are non-programmable

and without graphic displays.5. Diagrams are not drawn to scale. They are intended as aids only.

PART A


1. In the diagram, ABCD is a parallelogram withA , B , C , D in the first quadrant, as shown. What arethe coordinates of C ?

y

x

C

B (7, 3)

D (3, 7)

A (2, 3)

2. Mr. Matheson has four cards, numbered 1, 2, 3, 4. He gives one card each to Ben,Wendy, Riley, and Sara. Ben is not given the number 1. Wendy’s number is 1 greaterthan Riley’s number. Which number could Sara not have been given?

3. If 99!

101! − 99! = 1

n, determine the value of n.

(If m is a positive integer, then m! represents the product of the integers from 1 to m,inclusive. For example, 5! = 5(4)(3)(2)(1) = 120 and 99! = 99(98)(97) · · · (3)(2)(1).)

4. In the diagram, ABCDEF is a regular hexagonwith side length 4 and centre O. The line segmentperpendicular to OA and passing through A meetsOB extended at P . What is the area of

OAP ?

A B

C

D E

F

P

4

O

5. Each of the positive integers 2013 and 3210 has the following three properties:

(i) it is an integer between 1000 and 10 000,

(ii) its four digits are consecutive integers, and

(iii) it is divisible by 3.

In total, how many positive integers have these three properties?


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6. If p and q are positive integers, max( p, q ) is the maximum of p and q and min( p, q )is the minimum of p and q . For example, max(30, 40) = 40 and min(30, 40) = 30.Also, max(30, 30) = 30 and min(30, 30) = 30.

Determine the number of ordered pairs (x, y) that satisfy the equation

max(60, min(x, y)) = min(max(60, x), y)

where x and y are positive integers with x ≤ 100 and y ≤ 100.

PART B

For each question in Part B, your solution must be well organized and contain wordsof explanation or justification when appropriate. Marks are awarded for completeness,clarity, and style of presentation. A correct solution, poorly presented, will not earn fullmarks.

1. At Galbraith H.S., the lockers are arranged in banks of 20 lockers. Each bank of lockers consists of six columns of lockers; the first two columns in each bank consistof two larger lockers and the last four columns in each bank consist of four smallerlockers. The lockers are numbered consecutively starting at 1, moving down eachcolumn and then down the next column, and so on. The first twenty-one lockers andtheir locker numbers are shown in the diagram.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

(a) What is the sum of the locker numbers of the column of lockers that containthe number 24?

(b) The sum of the locker numbers for one column is 123. What are the lockernumbers in this column?

(c) The sum of the locker numbers for another column is 538. What are the lockernumbers in this column?

(d) Explain why 2013 cannot be a sum of any column of locker numbers.

2. (a) Expand and simplify fully the expression (a− 1)(6a2 − a− 1).(b) Determine all values of θ with 6cos3 θ− 7cos2 θ + 1 = 0 and−180◦ < θ


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3. If m and n are positive integers, an (m, n)-sequence is defined to be an infinitesequence x1, x2, x3, . . . of A’s and B ’s such that if xi = A for some positive integer i,then xi+m = B and if xi = B for some positive integer i, then xi+n = A. Forexample, ABABAB . . . is a (1, 1)-sequence.

(a) Determine all (2, 2)-sequences.

(b) Show that there are no (1, 2)-sequences.

(c) For every positive integer r, show that if there exists an (m, n)-sequence, thenthere exists an (rm,rn)-sequence.

(d) Determine all pairs (m, n) for which there is an (m, n)-sequence.


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www.cemc.uwaterloo.ca

CanadianSenior

Mathematics ContestTuesday, November 20, 2012


Wednesday, November 21, 2012 (outside of North America and South America)



PART A


2. Enter the answer in the appropriate box in the answer booklet.For these questions, full marks will be given for a correct answer which is placed in the box.

Part marks will be awarded only if relevant work is shown in the space provided in theanswer booklet.

PART B




NOTES:

The questions in each part are arranged roughly in order of increasing difficulty.

The early problems in Part B are likely easier than the later problems in Part A.


answer booklet.



published on the Web site, http://www.cemc.uwaterloo.ca. In addition, the name, grade, school

and location, and score of some students may be shared with other mathematical organizations

for other recognition opportunities.


14/20


NOTE: 1. Please read the instructions on the front cover of this booklet.2. Write solutions in the answer booklet provided.3. 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 . . .

or 4.646 . . ..4. Calculators are permitted, provided they are non-programmable

and without graphic displays.5. Diagrams are not drawn to scale. They are intended as aids only.

PART A


1. Figure ABCDEF has AB = 8, BC = 15, andEF = 5, as shown. Determine the perimeterof ABCDEF . A

B C

D E

F

15

8

5

2. There are three distinct real numbers a, b and c that are solutions of the equationx3 − 4x = 0. What is the value of the product abc?

3. If 3x = 320

·320

·318 + 319

·320

·319 + 318

·321

·319, determine the value of x.

4. Three boxes each contain an equal number of hockey pucks. Each puck is either blackor gold. All 40 of the black pucks and exactly 1

7 of the gold pucks are contained in

one of the three boxes. Determine the total number of gold hockey pucks.

5. In the diagram, square PQRS has sidelength 25, Q is located at (0, 7), and R is onthe x-axis. The square is rotated clockwiseabout R until S lies above the x-axis on theline with equation x = 39. What are the newcoordinates of P after this rotation?

x R

Q

P

S

y

x = 39

O

6. Lynne is tiling her long and narrow rectangular front hall. The hall is 2 tiles wideand 13 tiles long. She is going to use exactly 11 black tiles and exactly 15 whitetiles. Determine the number of distinct ways of tiling the hall so that no two blacktiles are adjacent (that is, share an edge).


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PART B


1. In each diagram shown in this problem, the number on theline connecting two circles is the sum of the two numbersin these two circles. An example of a completed diagram isshown to the right.

2

5

97

3 14

(a) What is the value of x? 413

10 x

(b) With justification, determinethe value of y.

y y

w

3w48

(c) With justification, determinethe values of p, q and r.

p

q

r

1318

3

2. Consider the equation x2 − 2y2 = 1, which we label 1. There are many pairs of positive integers (x, y) that satisfy equation 1.

(a) Determine a pair of positive integers (x, y) with x ≤ 5 that satisfies equation 1.(b) Determine a pair of positive integers (u, v) such that

(3 + 2√

2)2 = u + v√

2

and show that (u, v) satisfies equation 1.(c) Suppose that (a, b) is a pair of positive integers that satisfies equation 1.

Suppose also that (c, d) is a pair of positive integers such that

(a + b√

2)(3 + 2√

2) = c + d√

2

Show that (c, d) satisfies equation 1.(d) Determine a pair of positive integers (x, y) with y > 100 that satisfies equation 1.


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3. (a) Right-angled P QR has ∠P QR = 90◦, P Q = 6 and QR = 8. If M is themidpoint of QR and N is the midpoint of P Q, determine the lengths of themedians P M and RN .

(b) DEF has two medians of equal length. Prove that DEF is isosceles.(c) ABC has its vertices on a circle of radius r. If the lengths of two of the

medians of ABC are equal to r, determine the side lengths of ABC .


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www.cemc.uwaterloo.ca

CanadianSenior

Mathematics ContestTuesday, November 22, 2011


Wednesday, November 23, 2011(outside of North America and South America)



PART A



PART B


2. Finished solutions must be written in the appropriate location in the answerbooklet. Rough work should be done separately. If you require extra pages for your

finished solutions, they will be supplied by your supervising teacher. Insert these pages intoyour answer booklet. Be sure to write your name, school name and question number on anyinserted pages.


NOTES:


answer booklet.

The names of some top-scoring students will be published on the CEMC website,

http://www.cemc.uwaterloo.ca.


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NOTE: 1. Please read the instructions on the front cover of this booklet.2. Write solutions in the answer booklet provided.3. All calculations and answers should be expressed as exact numbers

such as 4π, 2 +√

7, etc., rather than as 12.566 . . . or 4.646 . . ..

4. Calculators are permitted, provided they are non-programmableand without graphic displays.

5. Diagrams are not drawn to scale. They are intended as aids only.

PART A


1. Determine the value of 241 + 12

+ 1

22 +

1

23 +

1

24.

2. Four years ago, Daryl was three times as old as Joe was.In five years, Daryl will be twice as old as Joe will be.How old is Daryl now?

3. A die is a cube with its faces numbered 1 through 6. One red die and one blue dieare rolled. The sum of the numbers on the top two faces is determined. What is theprobability that this sum is a perfect square?

4. Determine the number of positive divisors of 18 800 that are divisible by 235.

5. In the diagram, the circle has centre O. OF is

perpendicular to DC at F and is perpendicular to ABat E . If AB = 8, DC = 6 and EF = 1, determinethe radius of the circle.

O

A B

C D

E

F

6. In a magic square, the numbers in each row,the numbers in each column, and the numberson each diagonal have the same sum. Giventhe magic square shown with a, b, c, x , y , z > 0,determine the product xyz in terms of a, b and c.

log a log b

log c

log x

log y

log z

p

q r


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PART B


1. The parabola with equation y = 25−x2 intersects thex-axis at points A and B, as shown.

(a) Determine the length of AB.

(b) Rectangle ABCD is formed as shown withC and D below the x-axis and BD = 26.Determine the length of BC .

(c) If CD is extended in both directions, it meetsthe parabola at points E and F . Determinethe length of EF .

y

A B

25

C D

x

2. (a) First, determine two positive integers x and y with 2x + 11y

3x + 4y = 1.

Now, let u and v be two positive rational numbers with u < v.

If we write u and v as fractions u = a

b and v =

c

d, not necessarily in lowest

terms and with a,b, c, d positive integers, then the fraction a + c

b + d is called a mediant

of u and v. Since u and v can be written in many different forms, there are manydifferent mediants of u and v.

In (a), you showed that 1 is a mediant of 23

and 114

.

Also, 2 is a mediant of 2

3 and

11

4 because

2

3 =

6

9 and

11

4 =

44

16 and

6 + 44

9 + 16 = 2.

(b) Prove that the average of u and v , namely 12

(u + v), is a mediant of u and v.

(c) Prove that every mediant, m, of u and v satisfies u < m < v.

3. Suppose that n

≥3. A sequence a1, a2, a3, . . . , an of n integers, the first m of which

are equal to −1 and the remaining p = n− m of which are equal to 1, is called anMP sequence .

(a) The sequence −1,−1, 1, 1, 1 is the MP sequence a1, a2, a3, a4, a5 with m = 2and p = 3. Consider all of the possible products aia jak (with i < j < k) thatcan be calculated using the terms from this sequence. Determine how many of these products are equal to 1.

(b) Consider all of the products aia jak (with i < j < k) that can be calculatedusing the terms from an MP sequence a1, a2, a3, . . . , an. Determine the numberof pairs (m, p) with 1 ≤ m ≤ p ≤ 1000 and m + p ≥ 3 for which exactly half of these products are equal to 1.


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CSMC - Problems (2015 - 2011).pdf

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