AMO 2027 Grades 6-7 SET 1 - Self-study

Bryan bought an equal number of books and magazines for 120 dollars. A magazine cost 6 dollars and a book cost 3 dollars more than a magazine. How many books did he buy?

In the diagram below, ABCD is a square, ∠CDE = 72° and ∠CED = 54°. Find the angle ∠EAD.

A sociologist chose 300 students at random from each of two schools and asked each student how many siblings he or she has. The results are shown in the table below.

Students’ Sibling Survey

| Number of siblings | Lincoln School | Washington School |
|---|---|---|
0 | 120 | 140
1 | 80 | 110
2 | 60 | 30
3 | 30 | 10
4 | 10 | 10

There are a total of 2,400 students at Lincoln School and 3,300 students at Washington School. What is the median number of siblings for all the students surveyed?

The figure shows a rectangle PRSU with a line QT which divides the rectangle into two squares.
How many right-angled triangles can be drawn using any three points P, Q, R, S, T, U as corners?

After adding the digit ‘4’ on the right of a 4-digit number, a new 5-digit number is formed. The difference between these two numbers is 9787. What is the value of the original 4-digit number?

a) The Walter Corporation would like to donate 6 computers and 10 printers to local schools. The corporation would like to make sure that each school receives the same set of computers and printers, with none left over. What is the greatest number of schools that The Walter Corporation can donate to?

b) Lisa has 12 forks and 15 knives to place in cutlery holders at the fast food restaurant where she works. She wants to distribute them equally, with no forks or knives left over. What is the greatest number of cutlery holders Lisa can stock?

What is the maximum number of possible points of intersection of a circle and a triangle?

A car went a distance of 90 km at a steady speed and returned along the same route at half that speed. The time needed for the whole round trip was four hours and a half. Find the two speeds.

Carl paid 108 dollars for 28 books and magazines altogether. He misplaced 4 books. The number of books left was 0.2 the number of magazines bought. Each book cost 3 dollars more than a magazine. How much more did he pay for the magazines than the books?

The following figures are made up of squares and dots.

a) Calculate the number of dots in figure 10.

b) Calculate the number of dots in figure 100.

c) Which figure has 132 dots?

d) Which figure has 288 dots?

Given that E, F, G, H are midpoints of AB, BC, CD, DA, respectively. Find the area of the square IJKL if AB = 10cm.

The figure below shows a square ABCD, which is divided into 4 identical rectangles. The perimeter of each rectangle is 50cm. Find the perimeter (in cm) of the square ABCD.

Seven friends met at a party. Each of them shook hands with everyone else in the party exactly once. How many handshakes happened?

a) The first number is three times the second number. The sum of their reciprocals is 4. What is the value of the second number?

b) $\frac{2}{9}$ of the people on a restaurant are adults. If there are 95 more children than adults, how many children are there in the restaurant?

c) Susan jogged from home to the sport complex at a constant speed of 6 km/h. 10 minutes later, her sister, Susie, started riding a bicycle along the same route toward the sports complex at a constant speed of 18 km/h. If they arrived at sports complex at the same time, how many seconds does it take Susan to jog from home to the sport complex?

Sage has a walking speed of 300 feet per minute. On the way to gate 14C at the airport, Sage has the option of using a moving sidewalk. By simply standing on the sidewalk, it would take 4 minutes to get to the gate that is 800 feet away.

a) How much time will it take Sage to walk the distance to the gate without using the moving sidewalk?

b) How much time will it take Sage to get to the gate by walking on the moving sidewalk?

c) After traveling 200 feet (by standing on the sidewalk), Sage notices a Moonbucks, and turns around on the moving sidewalk. How long will it take Sage to get back to the beginning of the moving sidewalk, walking in the opposite direction? Assume the sidewalk is empty of other travelers.