AMO 2027 Grade 3 SET 5 - Self-study

Jack and his father’s weight add up to 100 kg. If Jack loses 4 kg, he will be exactly half of his father’s weight. What is the difference between their weights, in kg?

**Editorial note:** Find the difference before Jack loses weight; his father’s weight stays unchanged. Choose one option.

Zack, Tom and Jim each draw 3 cards from 9 cards numbered 1, 2, 3, …, 9.

Zack: The product of all my numbers is 35.

Tom: The product of all my numbers is 96.

Jim: The sum of my numbers is 16.

What is the second largest number of Jim?

**Editorial note:** The nine distinct cards are shared without replacement: each card goes to exactly one boy. Enter only the number, without units.

A basket contains an equal number of balls and pebbles. They are either white or black. The number of white balls is 30 and the total number of black items is 30. If the difference between the number of black and white balls is 20, find the number of white pebbles.

**Editorial note:** Enter only the number, without units.

The product of all the digits of a three-digit number is 12. How many such three-digit numbers are there?



Choose one option.

If the cells are added in the same pattern, which row will have the number 25?

**Editorial note:** Keep the printed fifth row exactly as shown. For this exercise, continue after 35 with 34, 33, 32, …, and put one more cell in each new row. Count the top row as row 1. Choose one option.

Tom scored one-third of his total score in English and two-fourths in Maths. His remaining 45 points were in French. What was his total score?



Enter one whole number, without units.

From a book with pages numbered 1 to 300, all pages whose numbers contain the digit 2 are removed. How many pages remain?

Use the source counting model: remove individual numbered pages; do not additionally remove the page printed on the reverse side of a physical sheet.



Choose one option.

Solve: $$100+90-80-70+60+50-40-30+20+10$$

**Editorial note:** Choose one option.

A car travels 44 km using 4 litres of petrol. Tom starts with 6 litres and uses one-third of it on Monday. He travels another 22 km on Tuesday. How many kilometres can the car travel with the petrol left? Assume the same fuel use per kilometre throughout and no extra petrol is added.



Enter one whole number, without units.

The numbers $20,21,22,\ldots,63,64$ are written together as the 90-digit number $2021222324\ldots626364$. All the odd digits are removed. How many digits remain? Remove individual digits 1, 3, 5, 7 and 9, not whole numbers that contain them.



Enter one whole number, without units.

An eraser costs 5 dollars less than a pencil. If 5 pencils and 5 erasers cost 40 dollars, what is the price of 2 pencils?



Choose one option.

A 5-digit number is a palindrome, meaning it reads the same forward and backward. It satisfies these conditions:
- The sum of the first three digits is 14.
- The third digit is half of the second digit.
- The sum of the second, third and fourth digits is 15.
Find the number.

Choose one option.

Three students each pick one ball from a bag containing three balls of different colours. How many different assignments are possible? Each ball is taken once, without replacement; students are distinct.



Choose one option.

$\overline{A6B}$ is a three-digit odd multiple of 15. If its digit sum is 15, find $A\times B$. The bar denotes the number with digits A, 6, B.



Choose one option.

A book has 250 pages. How many digits are used to print the page numbers of the book?

Enter one whole number, without units.