Prepare for AMO 2019 – Grade 8,9- Đề luyện thi AMO 2019 – Khối 8,9

Table Of Contents

Prepare for AMO 2019 – Grade 8,9

Week 1

Exercise 1. The below diagram shows an equilateral triangle and a square of side length 2 joined along an edge. What is the area of the shaded triangle?

Exercise 2. In ΔPQR, U is a point on PR, S is a point on PQ, T is a point on QR with US || RQ and UT || PQ. The area of ΔPSU is 120 cm2 and the area of ΔTUR is 270 cm2. What is the area of ΔQST?

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

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

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

Exercise 5. How many of the first 2019 numbers in the sequence 101, 1001, 10001, 100001,… are divisible by 101?

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.

Exercise 8. Let n be a positive integer and a be 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 have B blood type, 70 have O blood type and 15 have type AB blood. If a person from this group is selected at random, what is the probability that this person has O blood type?

Exercise 10. A box contains 5 chips, numbered 1, 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 is 3?

Exercise 11. A number m is randomly selected from the set {11, 13, 15, 17,19}, and a number n is randomly selected from {1999, 2000, 2001,….2018}. What is the probability that the unit digit of mn is 1 ?

Week 2

Exercise 1. A cylinder has radius 12 and height 30. The top circular face of the cylinder is the base of a cone and the centre of the bottom circular base of the cylinder is the vertex of the cone. A sphere is placed inside so that it touches the cone, the base of the cylinder and the side of the cylinder as shown. Which of the following is closest to the radius of the sphere?
(A) 4.84        (B) 4.74
(C) 4.64        (D) 4.54
(E) 4.44

Exercise 2. ABCDEFGH is a cube having a side length of 2. P is the midpoint of EF, as shown.

What is the area of APB?

Exercise 3. The pattern of figures

is repeated in the sequence

What is the 2019th figures in the sequence?

Exercise 4. The 50th term in the sequence 5, 6x, 7x2, 8x3. 9x4 , …

Exercise 5. A data set contains 10 whole numbers. The smallest number is 2 and the largest number is 7. The mean, median and mode of the data set are all equal to 3. List the 10 numbers in ascending order (i.e. from the smallest to the largest).

Exercise 6. In a set of five numbers, the average of two of the numbers is 12 and the average of the other three numbers is 7. What is the average of all five numbers?

Exercise 7. On Monday, 10% of the students at Dunkley S.S. were absent and 90% were present. On Tuesday, 10% of those who were absent on Monday were present and the rest of those absent on Monday were still absent. Also, 10% of those who were present on Monday were absent and the rest of those present on Monday were still present. What percentage of the students at Dunkley S.S. were present on Tuesday?

Exercise 8. Find the next term of the following sequence: 1, 2, 5, 8,15, 28, 51, …

Exercise 9. The mean average of a set of six numbers is 10. If the number 25 is removed from the set, what is the mean of the remaining numbers?

Exercise 10. The regular price for a bicycle is $320. The bicycle is on sale for 20% off. The regular price for a helmet is $80. The helmet is on sale for 10% off. If Sandra bought both items on sale, what is her percentage savings on the total purchase?

Exercise 11. Laura paid $17 for a pair of jeans. The ticketed price was 20% off the original price plus the sign on the rack said, ”Take an additional 15% off the ticketed price”. What was the original price of the jeans?

Week 3

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

Exercise 2. In the multiplication shown, each of P,Q. R, S and T are a digit.

The value of P + Q + R+5+T is?

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

What is the value of X + Y + Z?

Exercise 4. In the diagram, M is the midpoint of YZ, the measure of ∠X AJZ = 30°, and the measure of ∠XYZ = 15°. What is the measure of ∠X ZY?

Exercise 5. In the diagram, OMNP is a square, O is a centre of a circle and OM is a radius. Point L is on the circle such that LK is perpendicular to MN and K is a midpoint of segment MN. Find the measure of angle ∠OLN.

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.

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 sold 3 senior citizen tickets and 1 child ticket for a total of $38. The school took in $52 on the second day by selling 3 senior citizen tickets and 2 child tickets. Find the total price of a senior citizen ticket and a child ticket.

Exercise 10. Some students want to order shirts with their school logo. One company charges $9.65 per shirt plus a setup fee of $43. Another company charges $8.40 per 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 landed 3 arrows in the outer ring and 5 arrows in the bull’s-eye, for a total of 440 points. Jacob went second and got 1 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?

Week 4

Exercise 1. In a group of people, the ratio of women to men is 12: 8. The average age of women is 22 and that of men is 32. What is the average age of people in this group?

Exercise 2. If x = –30 is one of the solutions of the equation x2 + 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 integer x. What is the value of f(31) ?

Exercise 4. Find the value of b/a if the polynomial x5 – x4 + 5x3 + 2x2 – 2x + 10 is divisible by ax3 + b.

Exercise 5. The function f (x) has the property that

f (2x + 3) = 2f (x) + 3
for all x. If f (0) = 6, what is the value of f (9)

Exercise 6. A woman is baking some cakes for her children. She counts the number of baked cakes and thinks, “If each child receive 5 cakes, then I have to bake 3 more 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 1/2 x (P (2) + 7P (6)).

Exercise 8. An egg is thrown from a platform 100m in the air. From the time t in seconds when thrown until it hits the ground below, the egg’s height from the ground in metres follows the formula h = –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 first 5 seconds.

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?

Exercise 10. The functions f and g satisfy

f(x) + q(x) = 3x + 5,
f(x) g(x) = 5x +7.

for all values of x. Determine the value of  |2f(2) g (2) |.

Exercise 11. Lately, my wrist watch is running 2 minutes more slowly after 24 hours, and my alarm clock is running 15 minutes more slowly after 24 hours. 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 is 4 minutes and 15 seconds.” So, what time is it when I ask my mother about time?

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