AMO 2027 Grades 6-7 SET 3 - Self-study

Cate baked a total of 450 chocolate cookies and peanut cookies. After she gave away 0.6 of the chocolate cookies and 0.5 of the peanut cookies, she had an equal number of chocolate and peanut cookies. How many peanut cookies had she left?

The distance from here to the beach at Little Boar’s Head is 10 miles. If you walked there at 4 mph and returned jogging at 8 mph, how much time would the round trip take? What would your overall average speed be?

How many triangles are there in the figure below?

The width of a rectangle is $(3y+5)$ cm. The perimeter of the rectangle is $(24y+10)$ cm.

a) Find the length of the rectangle.

b) Find the area of the rectangle if $y=3$.

How many times do the hour hand and minute hand of a standard clock meet each other in 24 hours?


A. 23

B. 24

C. 25

D. 26

E. None of the above

Denelle’s Bath Shop sells bars of soap in boxes of 8 bars and bottles of soap in boxes of 12 bottles. An employee is surprised to discover that the shop sold the same number of bars and bottles last week. What is the smallest number of each type of soap that the shop could have sold?

6 identical solid cubes were placed in a tank as shown below. The side of each cube was 4cm. Water was then poured in until it reached the height of the cubes.

a) Find the total volume of the cubes.

b) Find the volume of water in the tank.

There are 500 lockers numbered 1 to 500. Also, there are 500 students coming to the locker room. The first student opens all the 500 lockers. The second student then closes every locker with an even number, i.e. 2, 4, 6,... The third student then ‘reverses’ every third locker, i.e. if the locker is open, he/she will close it; but if the locker is closed, he/she will open it. The fourth student then ‘reverses’ every fourth locker, and so on until the 500ᵗʰ student has ‘reversed’ the 500ᵗʰ locker. What is the locker with the largest number that will remain open in the end?

A total of 4440 students attended a concert. After $\frac15$ of the boys and 925 girls left, the ratio of the number of boys to the number of girls became 8 : 9. Find the ratio of the number of boys to the number of girls at the concert at first.

Find the next number in the following sequence:

4, 12, 8, 16, 12, 20, 16, 24, ...

How many positive integers n are there such that $\frac{3n-1}{n-2}$ is also a natural number?

$\frac19$ of the shirts sold at Peter’s shop are striped. $\frac58$ of the remainder are printed. The rest of the shirts are plain colored shirts. If Peter’s shop has 81 plain colored shirts, how many more printed shirts than plain colored shirts does the shop have?

Thomas has 9 dollars more than Tom. Alice has 20 dollars fewer than Tom. The total amount of money they have is 199 dollars. How much money does Thomas have?

A square is labelled anticlockwise ABCD. The point E is outside the square so that CDE is an equilateral triangle. Find angle ∠BED.

A sequence is created by beginning with and repeatedly turning the arrow 45° counter clockwise. The first terms of the sequence are
→, ↗, ↑, ↖, ←, ↙, ...
Determine the 2019th figure in the sequence.