AMO 2027 Grade 6 SET 2 - Self-study

What is the greatest three-digit integer divisible by 24, 28 and 32?

**Input:** Enter the integer only, without units.

In the repeating pattern ODAKOOODAKOOODAK…, how many letters O are written before the 1808th letter? “Before” excludes the 1808th position.

**Input:** Choose one option.

Theo’s football coaching starts at 6:30 AM, and his mother wants him to wake up at 6 AM to be on time for coaching. Currently Theo wakes up late. Theo promises to wake up 5 minutes earlier than he did the day before. If Theo woke up at 6:50 AM on a Sunday and keeps his promise every day, on what day will he first wake up at 6:00 AM, the requested on-time wake-up target? Leave your answer as 0001 for Monday, 0002 for Tuesday, …, and 0007 for Sunday.

**Input:** Enter the four-digit weekday code: Monday 0001, Tuesday 0002, Wednesday 0003, Thursday 0004, Friday 0005, Saturday 0006, Sunday 0007.

What is the difference between the sum of the multiples of 3 less than 200 and the sum of the multiples of 5 less than 250? Here multiples are positive; subtract the second sum from the first.

**Input:** Enter the integer only, without units.

How many 4-digit multiples of 37 are there in total?

**Input:** Choose one option.

Lucy is Sherry’s mother. Their current ages are two-digit whole numbers with the same digits in reverse order. In 13 years, Lucy will be twice as old as Sherry. How old will Lucy be when Sherry reaches Lucy’s current age?

**Input:** Enter the integer only, without units.

Find the sum of the digits of $93517+35179+51793+17935+79351$.

**Input:** Choose one option.

A football league has 10 teams. During the season, each of the 10 teams plays exactly 3 games with each of the other teams. What is the total number of games played?

**Input:** Choose one option.

For example, $2^5=2\times2\times2\times2\times2$. If $4096=a^b$, find the sum of all possible values of $a$, where $a$ is a positive integer and $b$ is a whole number greater than 1.

**Input:** Choose one option.

The mean of four consecutive odd numbers is 24. Find the sum of the middle two numbers.

**Input:** Enter the integer only, without units.

In the cryptarithm $DUCK+DUCK=GOOSE$, different letters represent different digits and repeated letters represent the same digit. What is the least possible value of the three-digit number $GOO$? Leading digits are nonzero.



**Input:** Enter the integer only, without units.

The points A, B, C, D and E are arranged as shown so that no three lie on a straight line. How many different triangles can be formed by connecting three of these points as vertices?



**Input:** Choose one option.

As shown, the overlap of rectangles ABCD and EFGH is also a rectangle. The area of ABCD is $174\text{ cm}^2$ and the area of EFGH is $52\text{ cm}^2$. $EF=13\text{ cm}$ and $BC=29\text{ cm}$. What is the area, in square centimeters, of the entire figure?



**Input:** Enter the integer only, without units.

Find the largest four-digit number that leaves a remainder of 3 when divided by each of 4, 7 and 13.

**Input:** Enter the integer only, without units.

Remi had a square garden with an area of 16 square meters. She extends the length of the garden by 3 meters and the width by 2 meters. What is the new perimeter, in meters, of the garden?

**Input:** Choose one option.