AMO 2027 Grade 5 SET 3 - Self-study

Three pencils and one marker cost 2.10 dollars. One pencil and five markers cost 6.30 dollars. What is the total cost of 14 pencils and 14 markers?

Enter the total number of dollars, without a currency symbol or units.

The total area of the shaded parts of the picture is $18\text{ cm}^2$. Find the area, in square centimetres, of triangle CAB.



The grid consists of equal squares. Choose one option.

Which of the following numbers has the most number of even prime factors?

Count repeated prime factors with their multiplicities, as in the source solution.

Choose one letter A–E.

A train travelling at 54 km/h passes a platform. A man is standing on the platform, and the train passes the man in 20 seconds. Find the length of the train, in metres.

Enter one whole number, without units.

Five natural numbers are chosen from 1 to 45 and sum to 45. Let B be the largest of the five numbers. What is the largest possible value of B?

Clarification for this edition: the five chosen numbers must be distinct (no repetitions). The source does not explicitly state this condition.

Enter one whole number, without units.

The figure below is made of three regular pentagons, each with side length 3 cm. What is the perimeter of the figure?



Choose one option.

Counting numbers here are the positive integers starting at 1, for example 1, 2, 3, 4, … . What is the average of the sum of the first 4 counting numbers and the sum of the first 5 positive even numbers?

Choose one option.

Amanda's score was twice Brian's score. Cassie scored 5 points less than Brian. Dora scored 10 points more than Amanda. Dora's score was 6 times Cassie's score. Whose score was 10?

Choose one option.

Dr. Hazma, Dr. Tan and Dr. Gupta created vaccines in their labs. Their vaccines showed 98%, 92% and 93% effectiveness respectively. What is the probability that all three are simultaneously effective? Assume independent trials. Round the percentage to two decimal places.

Choose one option.

In a magic square, every row, column and diagonal has the same sum. The odd numbers from 1 through 17 are placed in this square, each once. What number goes in the cell marked A?



Choose one option.

What is the value of $(888+666+444+111)-(555+333+222+777)$?

Choose one option.

There are 165 unit cubes arranged in 5 square layers with no space between the cubes as shown. The layers are 1 by 1, 3 by 3, 5 by 5, 7 by 7, and 9 by 9. If a sixth layer is placed on the bottom of the arrangement following the pattern, how many unit cubes are completely surrounded by six other unit cubes?

Follow the centred, aligned square-layer arrangement shown; “surrounded” means sharing each of the six faces with a neighbouring cube.



Choose one letter A–E.

How many perfect squares less than or equal to 200 are the sum of two consecutive whole numbers? Here whole numbers include 0.

Enter one whole number, without units.

How many digits does $N$ have if $N=2\times3\times4\times5\times12\times25$?

Choose one option.

Find $(115+114+113+112+111)-(101+102+103+104+105)$.

Choose one option.