AMO 2027 Grades 4-5 SET 33 - Self-study

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.

Robin thought of a five-digit number that begins with 8. Her number is the smallest such number divisible by 9, and all its digits are different. Find the last four digits of the number.

Enter exactly four digits, retaining any leading zero. Do not insert spaces.

Three fruit vendors make 3 fruit baskets in 3 minutes. How many fruit baskets will be ready if 10 fruit vendors work for 15 minutes?

Assume all vendors work independently at the same constant rate, with enough materials. Choose one option.

If 52 is added to two-fifths of a number, the result is three times the number. What is the number?

Choose one option.

Place the whole numbers 1 through 7 in the figure, one per square. The sum in the left column, the sum in the right column, and the sum on each diagonal must all be equal. What is the largest possible product of the numbers across the grey row?



Use each of the seven numbers exactly once. Choose one option.

A four-digit number $\overline{AABB}$ is a perfect square. What is its square root?

A and B are digits and A is not zero; the bar denotes a four-digit numeral. Enter its positive square root as one whole number.

Brenda multiplies a number by 3, adds 3, divides by 3, adds 6, and subtracts 3 from the result to get 9. What is six times the original number?

Choose one option.

Mangoes were bought and placed into baskets to be distributed in an orphanage. If each basket was filled with 13 mangoes, 54 mangoes were left. But when 16 mangoes were put in each basket, there was a shortage of 27 mangoes. How many mangoes were bought?

Enter one whole number, without units.

What is the greatest three-digit multiple of 19 that is divisible by 3?

Enter one whole number, without units.

In the pyramid below, each number is the sum of the two numbers immediately below it. For example, three 1s on the lowest layer give two 2s above, then one 4 above them. If a similar pyramid has five 1s on its lowest layer, what is the sum of the numbers on its topmost layer?



Choose one option.

The keys to nine rooms are all mixed up in a box. What is the least number of key trials needed to guarantee that every key can be matched to its room lock?

There are exactly nine distinct keys, each opening exactly one different lock. Count physical key-in-lock trials; a match known by deduction needs no trial. Find the worst-case guarantee, not a lucky outcome. Enter one whole number.

One straight line through the region inside a circle can cut it into two parts, as shown. If the circle is cut into 232 parts, what is the least number of straight lines that must be drawn through the region?



Lines are straight and may be placed freely; count regions inside the circle. Enter one whole number.

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.

Sarah and Rose have some pencils. Two-fifths of Sarah’s pencils and eight-ninths of Rose’s pencils are each 40. Who has more pencils, and by how many?

Choose one option.

The figure below is made of five identical rectangles. Its total area is $60\text{ cm}^2$. What is the perimeter, in centimetres, of one small rectangle?





Enter one whole number, without units.