AMO 2027 Grades 6-7 SET 9 - Self-study

The following figure is formed by two identical squares of area 12cm². The area of shaded portion is 16cm². What is the area of the non-shaded portion?

The three angle bisectors of triangle LMN meet at a point O as shown. Goven that ∠LNM = 68°, find ∠LOM?

The figure is made up of 2-cm cubes. Find the volume of the figure.

How many numbers n have the property that both n/2 and 2n are four-digits whole numbers?

A. 1499
B. 1500
C. 2499
D. 2500
E. None of the above

If I were to increase my cycling speed by 3mph, I calculate that it would take me 40 seconds less time to cover each mile. What is my current cycling speed?

Andrew and Patsy had $\$875$ altogether. Andrew had $\$415$ more than Patsy at first. Both of them went shopping and Andrew spent 1/3 of the amount Patsy spent. In the end, Patsy had 1/4 as much money left as Andrew. How much money did Andrew have in the end?

Jack bought a rectangular shape drawing paper measuring 42cm by 27cm. He wants to cut the drawing paper into as many square shape papers as possible with the largest length.


a) What is the length of each square?

b) How many square shape papers can Jack made?

Two circles form a friendly pair if they touch each other. How many friendly pairs are there in the following diagram?

If numbers are arranged in 3 rows A, B and C according to the following table, which row will contain the number 1000?
A 1, 6, 7, 12, 13, 18, 19, ...
B 2, 5, 8, 11, 14, 17, 20, ...
C 3, 4, 9, 10, 15, 16, 21, ...

Suppose

$$\frac mn=\frac{2}{1\times3}+\frac{2}{3\times5}+\frac{2}{5\times7}+\frac{2}{7\times9}+\cdots+\frac{2}{49\times51},$$

where m/n is a fraction in simplest form. What is the value of m × n?

Mrs. Tan made some tarts. 1/4 of them were lemon tarts and the rest were strawberry tarts. After 210 lemon tarts and 5/6 of strawberry tarts were sold, she still had 1/5 of the tarts left. How many tarts did Mrs. Tan make at first?

a) In John’s stamp collection, the number of Malaysia stamps was 0.8 that of Singapore stamps. The number of Australia stamps was 0.5 that of Malaysia stamps. If he had 120 Singapore stamps, how many stamps did he have altogether?

b) Johnson spends 0.6 of his money on a pen and 0.15 of his money on a movie ticket. If he has $\$13$ left, how much money does he have at first?

c) A blouse cost $\$4.50$ more than a shirt. A skirt cost $\$15.20$ more than the blouse. Sandy bought 6 of each item. She paid $\$451.20$ for them. How much did she pay for a skirt?

**Editorial input note:** Enter the numerical answers to (a), (b), and (c), in order, separated by semicolons with no spaces. Use dollars for (b) and (c), with two digits after the decimal point for (c); omit currency symbols and unit labels.

Ages of the First 12 United States Presidents at the Beginning of Their Terms in Office
President | Age (Years) | President | Ages (Years)
Washington | 57 | Jackson | 62
Adams | 62 | Van Buren | 55
Jefferson | 58 | Harrison | 68
Madison | 58 | Tyler | 51
Monroe | 59 | Polk | 50
Rooney | 58 | Taylor | 65
The table above lists the ages of the first 12 United States presidents when they began their first term in office. According to the table, what was the mean age, in years, of these presidents at the beginning of their terms? (Round your answer to the nearest tenth.)

George is trying to unlock the forgotten passcode for his luggage. The passcode consists of 3 digits, each of which is from 0 to 9.
He tried 923, only one digit is correct and correctly placed.
He tried 958, only one digit is correct but wrongly placed.
He tried 379, only two digits are both correct but wrongly placed.
He tried 142, none of the numbers is correct.
He tried 127, only one digit is correct but wrongly placed.
What is George’s passcode?

The digits 2, 4, 6 and 8 are each used once to create two 2-digit numbers. What is the largest possible difference between the two 2-digit numbers?