AMO 2027 Grades 4-5 SET 32 - Self-study

A box contains balls and bats. If three-sevenths of the items are balls and there are 32 bats in the box, what is the total number of balls and bats in the box?

Choose one option.

If $A+B=28$, $B+C=40$ and $A+C=36$, what is $5\times A+5\times B+5\times C$?

Choose one option.

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.

Orville has L-tiles, each made up of three $1\times1$ squares as shown. What is the greatest number of these L-tiles that can fit into a $14\times10$ rectangle without overlap? The tiles can be rotated to fit, but cannot be cut.



Enter one whole number, without units.

Four friends Kevin, Larry, Pam and Sam had four bags of 30, 32, 35 and 38 marbles. Sam had more marbles than Pam but did not have the greatest number. Kevin had the least number. Which two friends have a total of 70 marbles?

Choose one option.

Anthony has at most 100 mathematics homework exercises to finish during his summer break. He finished two-fifths in the first week and three-sevenths in the second week, leaving 12 exercises unfinished. How many exercises were assigned?

Both fractions refer to the total number originally assigned. Choose one option.

The following $4\times4\times4$ cube is painted red on all its faces. Find the number of smaller cubes with at most two faces painted.





Choose one option.

The largest natural number formed by the digits 4, 5, 0, 3 and the smallest number formed by using all of those digits exactly once are subtracted. A valid number does not start with 0 unless it is 0 itself (considered a one-digit number). What is the number formed by the first two digits of the result?

Use each of the four digits exactly once in each number; subtract the smaller number from the larger. Choose one option.

Ron and Vanda each thought of a two-digit number. The products of their digits were 24 and 35 respectively. The difference between their numbers was 37. What is the sum of their numbers?

Choose one option.

In a book of 100 pages, all the pages with digits that are factors of 5 are torn. How many pages are left now?

Number the pages 1 to 100 inclusive. Remove each numbered page independently if its number contains at least one digit that is a positive factor of 5. Enter one whole number.

Adam can paint a classroom in 6 hours. Ben can paint the same classroom in 12 hours. If they work together, each at their own rate, how many hours will they take to paint the classroom?

Enter one whole number, without units.

Scott, Emily and Jack chose a number. Emily said it was an even multiple of 13. Scott said it was divisible by 35. Jack said it was a four-digit number with digit sum 15. Find the number.

Enter one whole number, without units.

Gerald likes to create patterns with dots. The first four figures are shown below. How many dots will the 16th figure contain?



Continue the same diamond-shaped outer-layer growth shown in the figures. Enter one whole number.

Tom read a book with 2022 pages. How many digits were used to print the page numbers in the book?

The pages are numbered consecutively from 1 to 2022, without leading zeros. Enter one whole number.

The product of $k$ and 1260 is a square number. If $k$ is a whole number, what is its least possible value?

Here whole numbers mean 0, 1, 2, …, and a square number may be the square of any whole number. Enter one whole number.