AMO 2027 Grade 5 SET 2 - Self-study

How many digits are in the least positive multiple of 41 whose digits are all 1?

Enter one whole number, without units.

The sum of three consecutive whole numbers is 443 greater than the greatest of them. What is the least of the three numbers?

Choose one option.

The sum of 5 different non-zero whole numbers is 45. What is the largest possible value of the largest of these 5 numbers?

Enter one whole number, without units.

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.

Each position in a four-digit PIN code can be any digit from 0 to 9. How many possible codes are there? Leading zeros and repeated digits are allowed. The standalone source phrases this as the number of possible outcomes when guessing a four-digit ATM PIN; the mathematical conditions are the same.

Choose one option.

Ari, Barry and Carrie chose a number. Ari said it was a multiple of 3, 5 and 11. Barry said it was a multiple of 2 and 7. Carrie said it had 4 digits, one of which was 9, but none of which were 6. Find the last 3 digits of the number.

Choose one option.

Karen can paint a wall in 2.5 hours. Stephanie takes 1.25 hours to paint the same size wall. How many minutes will it take for Karen and Stephanie working together to paint the same wall?

Assume constant individual painting rates, with their rates adding when they work together.

Enter only the whole-number answer, without units.

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.

A $5\times5\times5$ cube is painted red on all its faces. Find the number of smaller $1\times1\times1$ cubes with at most two faces painted.

Enter only the whole-number answer, without units.

In the cryptarithm MINED + DENIM, different letters represent different digits. Find the last three digits of the largest possible sum. Both addends are five-digit numbers. For example, if the sum were 1234, write 234.



Enter exactly the last three digits, without spaces.

Azma took part in a gymnastics competition. When the ranking list arrived, no one had been disqualified. She was three places ahead of the first contestant in the lower half and three places behind the bronze medalist. First prize is gold, second is silver, and third is bronze. How many people competed?

Clarification for this edition: ranks have no ties; “three places” means a rank difference of 3; the upper and lower halves have equal numbers of contestants.

Choose one option.

What is the 2021st number in the sequence $7,9,11,13,\ldots$?

Continue the constant increase shown. Choose one option.

Each of the whole numbers 1 through 9 is written on 9 cards, one number per card. Three cards are dealt to each of the three players. The sum of Player One's cards is 15. The sum of Player Two's cards is 23. What is the highest single card Player Three could have?

Enter only the whole-number answer, without units.

The cube shown is constructed from identical rectangular slabs. How many slabs does the cube contain? The divisions in the original drawing indicate the same unit scale on all three directions; there are no gaps or overlaps.



Choose one option.

A dancer is allowed to step on only one of the following tiles: 1 blue, 1 green, 2 red, 2 orange and 3 grey. The grey tiles are sticky and make the dancer stop dancing. If the chance of getting stuck on the very first step is $m/n$, find $m+n$.

**Assumptions for this self-study exercise:** Each of the nine tiles is equally likely to be chosen, and $m/n$ is written in lowest terms with positive whole-number numerator and denominator. The original question does not state these conditions.

Enter one whole number, without units.