AMO 2027 Grades 8-9 SET 3 - Self-study

Compute the sum of all the numbers found in the figure.

A city is divided into four regions. The city council has decided that a new city hall, a new school, and a new movie theatre shall be built. The only condition is that the school and the movie theatre must not be in the same region. How many ways these four buildings be built in the city? (Ignore the time of construction)

a) A zoo has 3 times as many gorillas as tigers. There are 3 more tigers than zebras at the zoo. If $z$ represents the number of zebras, in terms of $z$, how many gorillas are in the zoo?

b) The total perimeter of an equilateral triangle and a square is $(38p+81)$ cm. If the length of a side of the equilateral triangle is $(2p+7)$ cm, find the length of a side of the square. Give your answer in terms of $p$.

Each of two boxes contains three chips numbered 1, 2, and 3. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?

The diagram below shows the flag of Sweden, which consists of a gold cross of uniform width against a solid blue background. The flag measures 3 feet 4 inches by 5 feet 4 inches, and the area of the gold cross is 30% of the area of the whole flag. Use this information to find the width of the gold cross.

Express the following numbers as product of prime numbers.

a) 512

b) 299

c) 1001

d) 2010

Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was 12.48 dollars, but in January her bill was 17.54 dollars because she used twice as much connect time as in December. What is the fixed monthly fee?

Use the information given to find the passcode of the padlock.

| Attempt | Information |
|---|---|
| 4 9 6 | One correct number and correctly placed. |
| 4 3 7 | Two correct numbers but wrongly placed. |
| 3 2 4 | One correct numbers and correctly placed. |
| 3 2 6 | Two correct numbers and correctly placed. |

The three passcode boxes are: ? ? ?

A tank is connected with 3 pipes P, Q, R. Pipes P and Q are inlet pipes and pipe R is an outlet pipe. Pipe P and Q fill the tank in 6 hours and 5 hours respectively and R empties the tank in 3 hours. If pipe P is opened at 7 PM and Q is opened at 8 PM and R is opened at 9 PM. What time does the cistern get filled?

Beatrice scored 76, 81, 72, and 93 marks for her four Mathematics tests. She wishes to improve her average mark by 3 marks. How many marks must she score for her next Mathematics test?

Three students A, B and C are asked to write down the height of a child, the circumference of a circle, the volume of a cup and the weight of a ball. Their responses are tabulated below:

| Student | Height of child (cm) | Circumference of circle (cm) | Volume of cup (cm³) | Weight of ball (g) |
|---|---:|---:|---:|---:|
| A | 90 | 22 | 250 | 510 |
| B | 70 | 21 | 245 | 510 |
| C | 80 | 22 | 250 | 520 |

If each student has only two correct responses, and the height of the child is $x$ cm, find the value of $x$.

Find the missing number.

a) $21,27,33,39,(\quad),51,(\quad),\ldots$

b) $0,3,9,18,30,(\quad),63,\ldots$

c) $1,4,9,(\quad),25,(\quad),\ldots$

d) $2,6,12,(\quad),(\quad),\ldots$

e) $11,88,16,80,21,72,(\quad),(\quad),\ldots$

Jack and Bill had 440 dollars altogether. Bill had 20 dollars more than Jack. When Jack spent $1/3$ the amount of money spent by Bill, he had 3 times as much money as Bill. How much money did Bill spend?

In the diagram, all triangles are equilateral. The total number of equilateral triangles of any size is

The computer selects a random positive integer from 1 to 10 to show on the screen. What is the probability that the computer shows a prime number?