AMO 2027 Grades 8-9 SET 7 - Self-study

Define the symbol $[x]$ to be the greatest integer less than or equal to $x$. For example: $[12.7]=12$, $[2]=2$ and $[-3.2]=-4$.

Compute the integer square value of the expression below.
$$\frac{[112000.111222]\times[300.456]}{[1200000.987654321]}$$

In the diagram, $ABC$ is a right triangle. Given that $DE$ is perpendicular to $AC$, $AB=5$ cm, $AC=13$ cm and $DE=3$ cm, find the area of the polygon $ABED$.

The sum of four consecutive integers is 2174. What are the integers?

**Editorial input format:** Enter all four integers in increasing order, separated by commas.

**Editorial input note:** Use no spaces before or after commas.

In how many ways can we place 8 identical rooks on a $8\times8$ chessboard such that no two rooks can capture each other ?

The sum of two numbers is 340. Its quotient and remainder are 7 and 20 respectively when one of them is divided by the other. Find the two numbers.

**Editorial input format:** Enter both numbers, separated by a comma.

**Editorial input note:** Enter the smaller number first, with a comma and no spaces between the two numbers.

10 buckets of water fill a tank. The capacity of each bucket is 9.5 liters. How many buckets will be needed to fill the same tank if the capacity of each bucket is 19 liters?

A rectangular block, with dimensions 4cm, 5cm and 6cm, is made up of cubes each with side length 1cm. If $1\text{ cm}^3$ cubes are removed from this larger rectangular block, what is the minimum number of these cubes that must be removed so that the resulting solid is itself a cube?

Find the missing number.

The expression below can be simplified to an integer. What is that integer?
$$9700\times\left(\frac{1}{1\times4\times7}+\frac{1}{4\times7\times10}+\frac{1}{7\times10\times13}+\cdots+\frac{1}{94\times97\times100}\right)$$

The Math team of a school consists of ten students. Nine of the ten students are Albert, Anthony, Bernard, Bob, Cindy, David, Edward, Harry and Jim. The tenth student has the same name as one of the other nine. A total number of 54 letters, including four vowels in capital letter, must be used to write the names of all the ten students. What is the name of the tenth student?

The number of English books in a library is 326 more than the number of Chinese books and 634 more than the number of Malay books. The average number of English, Chinese and Malay books in the library is 1080. Find the average number of Chinese and Malay books in the library.

The spinner below is divided into 5 equal parts. Spin the needle once. If it lands on a perfect square number flip a fair coin once. If it lands on a prime number flip the coin twice. What is the probability you would see at least one head?

When the mean, median, and mode of the list
$$10,2,5,2,4,2,x$$
are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$?

Larry’s 3-digit number satisfies the following conditions:
- The number is not a multiple of 3.
- Exactly one of the digits is a prime number.
- Another digit is a square number.
- The other digit is neither prime nor square.

What is the largest possible value of Larry’s number?

What is the total number of different ways that the blanks below can be filled so that the resulting four-digit number is a multiple of 11 ?
$$8\underline{\phantom{0}}6\underline{\phantom{0}}$$