AMO 2027 Grades 8-9 SET 6 - Self-study

Square $ABCD$, shown here, has side length 8 units and is divided into four congruent squares. One of these squares contains an inscribed circle, two other squares contain diagonals and the fourth square has perpendicular line segments drawn from the midpoints of adjacent sides to form a square in the interior. In square units, what is the total area of the shaded regions? Take $\pi=22/7$.

Find the smallest value of $\square$ and $\triangle$ in the equation below.
$$25\times\triangle\times18=\square\times15$$

Let $X=A+B+C+D$. If $A$, $B$, $C$, and $D$ are each randomly assigned a number from $\{2,3\}$, what is the probability that $X=10$?

How many triples of positive integers have sum 20 and product a multiple of 5?

The highest common factor of two whole numbers is 20. The lowest common multiple of these two whole numbers is 200. If the sum of these two whole numbers is 140, what is the difference of these two whole numbers?

Cindy leaves school at the same time every day. If she cycles at 20km/h, she arrives home at 4 : 30 in the afternoon. If she cycles at 10km/h, she arrives home at 5:15 in the afternoon. At what speed, in km/h, must she cycle to arrive home at 5:00 in the afternoon?

A television set has a 30-inch screen size, i.e., the length of the diagonal of the screen is 30 inches. The aspect ratio of the screen is $4:3$. Find the height of the screen. Leave your answer to the nearest whole number if necessary. (Note: The aspect ratio of the screen refers to the ratio of the width of the screen to its height.)

On Saturday afternoon, Armand sent $m$ text messages each hour for 5 hours, and Tyrone sent $p$ text messages each hour for 4 hours. Which of the following represents the total number of messages sent by Armand and Tyrone on Saturday afternoon?

The following pattern is made up of matchsticks. How many triangles are there when there are 101 matchsticks?

A cistern can be filled by pipe $A$ in 20 hours while it can be emptied by pipe $B$ in 30 hours.

a) If both pipes are opened simultaneously, find the time taken to fill the cistern.

b) If $A$ is opened first, after 10 hours, pipe $B$ is also opened. Find the total time taken to fill the whole tank.

A family consists of five members: grandmother, father, mother, two sons $X$ and $Y$. One day, the family is given two movie tickets. When discussing who will use the two tickets, the ideas from the five members are grandmother and mother (A), father and mother (B), father and grandmother (C), grandmother and $X$ (D), father and $Y$ (E). Eventually, the grandmother’s idea is followed and the ideas of the other members are partly satisfied. Which idea is given by grandmother?

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?

There are 11 red marbles, 9 blue marbles and 8 green marbles in a box. In how many ways can we select 3 marbles of different colors from the box ?

Dean scored a total of 252 points in 28 basketball games. Ruth played 10 fewer games than Dean. Her scoring average was 0.5 points per game higher than Dean’s scoring average. How many points, in total, did Ruth score?

**Masses (kilograms)**

| | | | | | | |
|---|---:|---:|---:|---:|---:|---:|
| Andrew | 2.4 | 2.5 | 3.6 | 3.1 | 2.5 | 2.7 |
| Maria | $x$ | 3.1 | 2.7 | 2.9 | 3.3 | 2.8 |

Andrew and Maria each collected six rocks, and the masses of the rocks were shown in the table above. The mean of the masses of the rocks Maria collected was 0.1 kilogram greater than the mean of the masses of the rocks Andrew collected. What was the value of $x$?