AMO 2027 Grades 8-9 SET 2 - Self-study

$MNPQ$ is a square formed by 3 identical rectangles. The perimeter of each rectangle is 240 cm. How long is $PQ$?

You throw three regular six-sided dice. What is the probability that you get one odd number and two even numbers?

A bag contains 20 candies: 4 chocolate, 6 mint and 10 butterscotch. Candies are removed randomly from the bag and eaten. What is the minimum number of candies that must be removed to be certain that at least two candies of each flavour have been eaten?

The pictures below are made up of squares and 6-sided shapes, each of which is added one shape at a time. Figure 1 has 17 lines, figure 2 has 20 lines and figure 3 has 25 lines. How many lines will there be in figure 8?

Find the remainder when $5+4^2+4^3+4^4+\cdots+4^{2015}+4^{2016}$ is divided by 21.

Three pipes A, B and C can fill a tank in 15 minutes, 20 minutes and 30 minutes respectively. The pipe C is closed in 6 minutes before the tank is filled. How much time will the pipes take to fill the tank?

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 $y=3$.

The sum of five consecutive integers is 75. What is the sum of the largest and smallest numbers of these five integers?

Albert, Bernard, Cindy and David look at a quadrilateral and say:
- Albert: It is a parallelogram.
- Bernard: It is a trapezoid.
- Cindy: It is a square.
- David: It is a rhombus.

Only one of the statements given by them is incorrect. Who gives incorrect statement?

Let $a$ and $b$ be two consecutive whole numbers. Which of the following is ALWAYS true about $a\times b$?

The same 20 contestants, on each of 3 days, answered 5 questions in order to win a prize. Each contestant received 1 point for each correct answer. The number of contestants receiving a given score on each day was shown in the table below.

**Number of Contestants by Score and Day**

| | 5 out 5 | 4 out 5 | 3 out 5 | 2 out 5 | 1 out 5 | 0 out 5 | Total |
|---|---|---|---|---|---|---|---|
| Day 1 | 2 | 3 | 4 | 6 | 2 | 3 | 20 |
| Day 2 | 2 | 3 | 5 | 5 | 4 | 1 | 20 |
| Day 3 | 3 | 3 | 4 | 5 | 3 | 2 | 20 |
| Total | 7 | 9 | 13 | 6 | 9 | 6 | 60 |

What was the mean score of the contestants on Day 1?

Four children met together: Adam, Bobby, Charley and Daniel. Adam shook hands with 3 of these children, Bobby shook hands with 2, and Charley shook hands with 1. How many of the children’s hands did David shake?

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

1. Find the total volume of the cubes.
2. Find the volume of water in the tank.

Teams A and B are playing a series of games. If the odds for either team to win any game are even and Team A must win two or Team B three games to win the series. What is the probability that Team A wins the series?

The product of 2860 and $m$ ($m$ ≠ $0$) is a square number. Find the smallest possible value of $m$?