AMO 2027 Grades 6-7 SET 24 - Self-study

An integer must be subtracted from −74 to give the result −59. Find the square of that integer.

**Input:** Choose one option.

Which of the following Venn diagrams represents soccer players, high school Math enthusiasts, and students?

Editorial interpretation: “high school Math enthusiasts” means high school students who are enthusiastic about mathematics. Use the general membership relationships, not a particular surveyed group.

**Input:** Choose one option.

Michael opens his favourite mathematics puzzle book and notes that the product of the page numbers facing him is 462. Find the sum of these two page numbers.

Editorial convention: treat the two page numbers as consecutive positive integers, as the source intends; disregard the left/right parity convention of ordinary book pagination.

**Input:** Choose one option.

Find the sum of all the whole numbers between 41 and 50 inclusive which are multiples of either 2 or 3 or 5.

**Input:** Choose one option.

Ananya wants to tile a floor that is 24 m by 40 m. There are two types of tiles: a square that is 2 m on a side and an L-shape as shown. The L-shaped tile can be turned over or rotated if needed. What is the least number of tiles Ananya needs to tile the floor completely?

All indicated tile lengths are in meters. Tiles cover the floor without gaps or overlaps.



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

Fedrick is twice as old as his sister Mandy. Mandy is 3 years older than their brother Ray. The sum of their three ages is 33. What is Ray’s age?

**Input:** Choose one option.

A survey of 40 students at Mathlete Academy found that 27 students play sports and 20 students play an instrument. If 8 students in the survey do not play a sport and do not play an instrument, how many students play both a sport and an instrument?

**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.

How many whole numbers between 41 and 50 inclusive are multiples of either 2 or 3 or 5?

**Input:** Choose one option.

Find the smallest integer which, when divided by 7, gives a remainder of 0, but when divided by 10 gives a remainder of 1.

Editorial clarification: in this self-study version, seek the smallest positive integer. Over all signed integers, the original wording has no smallest solution.

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

40 people are in a movie theatre which offers two snack options: popcorn and chips. 26 people like popcorn but do not like chips. 32 people like popcorn. Everyone in the theatre likes at least one of the two snack options. Find the number of people who like both.

**Input:** Choose one option.

Each number from 1 through 16 is written in exactly one box of the 4 × 4 grid. A path is formed by placing consecutive numbers in adjacent boxes horizontally or vertically, but not diagonally. Two numbers are shown. Find the smallest possible sum of the numbers in the starred (*) boxes.



**Input:** Choose one option.

Several people were surveyed for their preference for 4 drinks A, B, C and D. All the people surveyed are represented in the graph below. If the fraction of people who preferred A to those who preferred D is $\frac{m}{n}$, what is $m+n$?

Editorial clarification: take m and n to be positive integers with m/n in lowest terms. The original statement omits this condition. The vertical axis shows the number of people.



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

One light flashes every 7 minutes and another light flashes every 2 minutes. If both lights flash together at 1 PM, how many minutes after 3 PM will both lights flash together for the first time?

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

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.