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

Here is a pattern of triangles. How many black triangles would there be in the next diagram of the sequence?

In the diagram, the area of triangle ACE is 56cm², which is twice the area of the parallelogram ABDF. What is the area of the triangle CDF?
(The diagram is not drawn to scale)

A tank 64cm long, 46cm wide and 36cm high was 1/8 full of water. Water was then pumped out of it at a rate of 16cm³ per second. How long would it take to empty all the water in the tank?

In the figure below, ABCG and CDEF are 2 squares, BD = 63cm, GF = 9cm. Find the area of the shaded part.

The total cost of 8 pencils and 5 pens is $\$62$. The total cost of 3 pencils and 3 pens is $\$30$. Find the difference in cost between a pen and a pencil.

a) For health reasons, Amir wants to drink eight glasses of water a day. He’s already had six glasses. What fraction of eight glasses does Amir have left to drink?

b) Joe gave 1/2 of his sandwich to Peter at lunchtime, and ate 1/3 of it himself. How much of the sandwich did he have left?

c) Katie and her family ordered a pizza for dinner and ate 3/4 of it. The next day, Katie ate 1/2 of what was leftover for lunch. What fraction of the original pizza did Katie eat for lunch?

d) Molly needs 5/8 of a cup of diced onion for a recipe. After chopping all the onion she has, Molly has 3/5 of a cup of chopped onion. How much more chopped onion does she need?

e) John bought 2 pounds of butter to make cookies. If he used 1/2 of a pound for chocolate chip cookies, 1/8 of a pound for peanut butter cookies, and 2/3 of a pound for sugar cookies, what part of the original 2 pounds is left?

f) Virgil ate 3/7 of a chocolate chip cookie; Aristotle ate 1/3 of the same cookie. How much of the cookie is left?

Let N = 123456789101112...4344 be the 79-digit number obtained that is formed by writing the integers from 1 to 44 in order, one after the other. What is the remainder when N is divided by 45?

a) The average of four numbers is 66. Find the expression for the new average when two numbers x and 2x are added to it.

b) 5 years ago, Tina was 2p years old. Next year, Tina’s brother, Jack, will be 7 years older than Tina. Find their total age in 10 years’ time.

c) A zoo has 3 times as many gorillas as tigers. There are 3 more tigers than there are zebras at the zoo. If z represents the number of zebras, in terms of z, how many gorillas are in the zoo?

Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to average 50 miles per hour for the entire trip?

a) How many leap years are there from year 1899 to 2001?

b) Today is Monday. Which day will it be after 61 days?

c) January 1ˢᵗ, 2007 was Monday. What day of the week lies on January 1ˢᵗ, 2008?

**Editorial input note:** Enter (a), (b), and (c) in that order, using digits for (a) and full English weekday names for (b) and (c). Separate parts with a semicolon followed by one space; use no space before a semicolon.

What is the largest product that can be formed from using the digits 2, 3, 4 and 5, and one multiplication sign? You are only allowed to combine the digits to form two numbers, e.g. 2 × 345, but you are not allowed to use indices, e.g. 2³ × 4⁵ is not allowed.

The diagram shows some cubes of the same size stacked at a corner of a room. How many cubes are there altogether? (Note: The floor is horizontal and the two walls are vertical. There are no gaps or holes behind the visible cubes.)

Dan paid $\$85$ for 20 pizzas and hamburgers altogether. After his friends ate 5 pizzas, the number of pizzas left was 0.25 of the number of hamburgers bought. Each pizza cost $\$5$ more than a hamburger. How much more did he pay for the pizzas than the hamburgers?

The average cost of 2 mangoes and 5 pineapples is $\$2.30$. The average cost of each pineapple is $\$1.95$. The difference in cost between the 2 mangoes $\$1.25$. Find the cost of each mango.

**Editorial input note:** Enter the two prices in dollars, larger first, each with two digits after the decimal point. Separate them with a comma and no spaces; omit currency symbols.

Six numbers in increasing order are 2, 3, 5, x, y, 20. The median of the numbers is 8 and the mean is 10. What is the value of y?