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

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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 1

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 do not have any drafts out of
the blank spaces in the test paper.

For more information, visit our official pages: or https://www.
facebook.com/giaoductienphong/.

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

Exercise 3. In the below figure, we have AB= 13, BC = 15 and AC = 4. Find the area of the
rectangleBECD.

D C

B E

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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 5. Chips are drawn randomly one at
a time without replacement until the sum of the values drawn exceeds 4. 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. 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

Tuan

TRAINING COURSE FOR AMO 2019

PREPARE FOR AMO 2019 - GRADE 8-9

Week 3

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 do not have any drafts out of
the blank spaces in the test paper.

For more information, visit our official pages: or https://www.
facebook.com/giaoductienphong/.

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

1 X 2

− 8 Y

4 5

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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

× 4

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 6 7 5

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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?

Y M

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

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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 2 team to the Level 1 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 1 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 5 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

Tuan

ANSWER KEY

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

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

Exercise 7. 004 Exercise 8. 003 Exercise 9. 22

Exercise 10. 12 Exercise 11. 25

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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 4

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 do not have any drafts out of
the blank spaces in the test paper.

For more information, visit our official pages: or https://www.
facebook.com/giaoductienphong/.

Exercise 1. In a group of people, the ratio of women to men is12 : 8. 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) = 6 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) + 3

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 4 cakes, then there will be 3 remaining cakes for me." So, how many children does this
woman have?

Exercise 7. Given a polynomial P (x) of degree 4 with leading coefficient 1 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 2 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+ 7.
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

Tuan