Tài liệu ôn thi toán AMO lớp 8, 9 năm 2019 | Kỳ thi toán AMO 2019

In the first break time, each girl bought a cheese cake, each boy bought a coke (both the price of a cheese cake and a coke are integers). In the second break time, each girl bought a co[r]

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan TRAINING COURSE FOR AMO 2019

PREPARE FOR AMO 2019 - GRADE 8-9

Week

Introduction:This problem set is to help students get used to the problems appearing in AMO Students should finish this set in 30 minutes

The space after each question is intentionally left blank for students to draft In the real test, students not have any drafts out of the blank spaces in the test paper

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Exercise 1. The below diagram shows an equilateral triangle and a square of side length2joined along an edge What is the area of the shaded triangle?

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan

Exercise 2. In 4P QR,U is a point onP R,S is a point on P Q,T is a point on QRwithU S kRQ

and U T kP Q The area of4P SU is120 cm2 and the area of4T U R is270 cm2 What is the area of4QST?

120 270

P U R

Q

S

T

Exercise 3. In the below figure, we have AB= 13, BC = 15 and AC = Find the area of the rectangleBECD

D C

B E

A

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan

Exercise 4. Which of the following expressions is never a prime number whenpis a prime number?

(A)p2+ 16 (B)p2+ 24 (C)p2+ 26 (D) p2+ 46 (E) p2+ 96

Exercise 5. How many of the first2019 numbers in the sequence 101,1001,10001,100001, are divisible by101?

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan

Exercise 6. Calculate the following sum

100−98 + 96−94 + 92−90 +· · ·+ 8−6 + 4−2

Exercise 7. Calculate the following sum

12−22+ 32−42+ .−20182+ 20192

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan

Exercise 8. Letn be a positive integer andabe a real number, calculate the following sums

1 +a+a2+a3+ .+an

Exercise 9. The blood groups of 200 people is distributed as follows: 50 have type A blood, 65

haveB blood type,70haveO blood type and15have typeAB blood If a person from this group is selected at random, what is the probability that this person hasO blood type?

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan

Exercise 10. A box contains5chips, numbered1,2,3,4, and Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds What is the probability that the number of draws is3?

Exercise 11. A number mis randomly selected from the set{11,13,15,17,19}, and a numbern is randomly selected from{1999,2000,2001, ,2018} What is the probability that the unit digit of

mn is1?

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan ANSWER KEY

Exercise 1. Exercise 2. 180 Exercise 3. 108

Exercise 4. C Exercise 5. 505 Exercise 6. 50

Exercise 7. 2039190

Exercise 8. a

n+1−1

a−1

Exercise 9. 0.35

Exercise 10. 15 Exercise 11. 0.4

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

PREPARE FOR AMO 2019 - GRADE 8-9 Week

Exercise 1. In the subtraction shown,X andY are digits What is the value ofX+Y?

1 X

− Y

4

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan Exercise 2. In the multiplication shown, each ofP, Q, R, SandT are a digit The value ofP +Q+ R+S+T is?

P Q R S T

×

4 P Q R S T

Exercise 3. In the addition shown, each of X, Y and Z represents a digit What is the value of

X+Y +Z?

X Y Z

+ X Y Z

Y Z

1

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan Exercise 4. In the diagram, M is the midpoint of Y Z, the measure of ∠XM Z = 30◦, and the measure of∠XY Z = 15◦ What is the measure of ∠XZY?

Z

Y M

X

Exercise 5. In the diagram,OM N P is a square,O is a centre of a circle andOM is a radius Point

Lis on the circle such thatLK is perpendicular toM N andK is a midpoint of segmentM N Find the measure of angleOLN[

O M

N P

K L

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan Exercise 6. Ms Forest likes to let her students choose who their partners will be; however, no pair of students may work together more than seven class periods in a row Adam and Baxter have studied together seven class periods in a row Carter and Dennis have worked together three class periods in a row Carter does not want to work with Adam Who should be assigned to work with Baxter? In your answer sheet, write 001 for Adam, 002 for Baxter, 003 for Carter, 004 for Dennis

Exercise 7. The high school math department needs to appoint a new chairperson, which will be based on seniority Ms West has less seniority than Mr Temple, but more than Ms Brody Mr Rhodes has more seniority than Ms West, but less than Mr Temple Mr Temple doesn’t want the job Who will be the new math department chairperson? In your answer sheet, write 001 for Ms West, 002 for Mr Temple, 003 for Ms Brody, 004 for Mr Rhodes

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan Exercise 8. As they prepare for the state championships, one gymnast must be moved from the Level team to the Level team The coaches will move the gymnast who has won the biggest prize and who has the most experience In the last competition, Roberta win a bronze medal and has competed seven times before Jamie has won a silver medal and has competed fewer times than Roberta Beth has won a higher medal than Jamie and has competed more times than Roberta Michele has won a bronze medal, and it is her third time competing Who will be moved to the Level team? In your answer sheet, write 001 for Roberta, 002 for Jamie, 003 for Beth, 004 for Michele

Exercise 9. The school that Stefan went to was selling tickets to a choral performance On the first day of ticket sales the school sold3 senior citizen tickets and1child ticket for a total of $38 The school took in$52on the second day by selling3senior citizen tickets and2child tickets Find the total price of a senior citizen ticket and a child ticket

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan Exercise 10. Some students want to order shirts with their school logo One company charges

$9.65per shirt plus a setup fee of$43 Another company charges $8.40per shirt plus a$58 fee For what number of shirts would the cost be the same?

Exercise 11. Monica and Jacob decided to shoot arrows at a simple target with a large outer ring and a smaller bull’s-eye Monica went first and landed3 arrows in the outer ring and arrows in the bull’s-eye, for a total of440 points Jacob went second and got1 arrow in the outer ring and

4 arrows in the bull’s-eye, earning a total of 317 points How many points was the outer ring is worth?

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Exercise 1. 10 Exercise 2. 14 Exercise 3. 15

Exercise 4. 75◦ Exercise 5. 135◦ Exercise 6. 003

Exercise 10. 12 Exercise 11. 25

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

PREPARE FOR AMO 2019 - GRADE 8-9 Week

Exercise 1. In a group of people, the ratio of women to men is12 : The average age of women is22and that of men is32 What is the average age of people in this group?

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan Exercise 2. If x=−30 is one of the solutions of the equationx2+ 12x+c= 0, what is the other solution of this equation ?

Exercise 3. The function f has the properties that f(1) = and f(2x+ 1) = 3f(x) for every integerx What is the value off(31)?

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan Exercise 4. Find the value of ab if the polynomial x5−x4+ 5x3+ 2x2−2x+ 10 is divisible by ax3+b

Exercise 5. The functionf(x)has the property that

f(2x+ 3) = 2f(x) +

for allx Iff(0) = 6, what is the value of f(9)

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan Exercise 6. A woman is baking some cakes for her children She counts the number of baked cakes and thinks, "If each child receive5cakes, then I have to bake3more cakes; but if each child receive cakes, then there will be remaining cakes for me." So, how many children does this woman have?

Exercise 7. Given a polynomial P (x) of degree with leading coefficient such that P(1) = 3, P (3) = 11,P (5) = 27 Find the value of 12 ×(P (−2) + 7P (6))

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan Exercise 8. An egg is thrown from a platform 100m in the air From the timet in seconds when thrown until it hits the ground below, the egg’s height from the ground in metres follows the formulah=−5t2+ 30t+ 100 Assume that air resistance is negligible Find the difference between the maximum height and the minimum height of the egg in the first5seconds

Exercise 9. A class has 30 students, most of them are boys One night, all of them went to the cinema In the first break time, each girl bought a cheese cake, each boy bought a coke (both the price of a cheese cake and a coke are integers) In the second break time, each girl bought a coke, each boy bought a cheese cake They realized that in the second break, they paid dollars less than in the first break How many boys does the class have?

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Tuan Exercise 10. The functionsf andq satisfy

f(x) +g(x) = 3x+ 5, f(x)−g(x) = 5x+ for all values ofx Determine the value of|2f(2)g(−2)|

Exercise 11. Lately, my wrist watch is running 56 minutes more slowly after24hours, and my alarm clock is running112 minutes more slowly after24hours At the midday of Monday, I adjust my wrist watch and clock A week later, I ask my mother what time it is Mom said, "I don’t know, but the difference in time between your watch and your clock is4minutes and15seconds." So, what time is it when I ask my mother about time?

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Instructors:

Dr T ran

Nam

Dung,

Dr L

e Minh

Exercise 4. Exercise 5. 33 Exercise 6.

Exercise 10. 28 Exercise 11. 9pm