AMO 2027 Grades 8-9 SET 4 - Self-study

The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$.

The rectangle shown below has been broken into four smaller rectangles. The areas of three of the smaller rectangles are shown in the diagram. Find the area of the fourth one.

Find the next term of the following sequence: $2,12,1112,3112,132112,\ldots$

Solve the equations:

a) $3x+15=3-4x$

b) $9x=\frac{2x}{5}+1$

c) $3x-4=11$

d) $9x+2=\frac34(2x+11)$

Find the LCM of

a) 9, 16, 24.

b) 8, 36, 60.

There are five people sitting at a round table. Each of them says: “Two people sitting next to me are all liars.” It is given that liars never tell the truth and people who are not liars always tell the truth. How many liars are there?

**Ages of the First 12 United States Presidents at the Beginning of Their Terms in Office**

| President | Age (Years) | President | Ages (Years) |
|---|---:|---|---:|
| Washington | 57 | Jackson | 62 |
| Adams | 62 | Van Buren | 55 |
| Jefferson | 58 | Harrison | 68 |
| Madison | 58 | Tyler | 51 |
| Monroe | 59 | Polk | 50 |
| Rooney | 58 | Taylor | 65 |

The table above lists the ages of the first 12 United States presidents when they began their first term in office. According to the table, what was the mean age, in years, of these presidents at the beginning of their terms? (Round your answer to the nearest tenth.)

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

An item in the store is discounted by 40%. The item is then placed on sale for 75% off of the lower price. By how many percent does the item’s regular price decrease after these two discounts are applied?

A cistern is connected with three inlet pipes A, B and C. Pipe A can fill the tank in 4 hours, pipe B in 8 hours, and C in 10 hours. All the pipes are opened simultaneously at initia. After $X$ hours, A and B closed and remaining tank is filled by pipe C alone. Find the value of $X$ if the total time taken to fill tank is 2 hours 30 minutes.

Tom writes all the natural numbers not exceeding 2021 whose digits are all even in ascending order: $2,4,6,8,20,22,24,26,28,\ldots$. How many numbers in total does Tom write?

A point is chosen at random within the square in the coordinate plane whose vertices are $(0,0)$, $(2020,0)$, $(2020,2020)$, and $(0,2020)$. The probability that the point is within $d$ units of a lattice point is $1/2$. (A point $(x,y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth?

In the diagram, $\triangle PQR$ is isosceles with $PQ=PR$ and $\triangle PRS$ is isosceles with $PS=SR=x$. Also, the perimeter of $\triangle PQR$ is 22, the perimeter of $\triangle PRS$ is 22 and the perimeter of $PQRS$ is 24. What is the value of $x$?

In a school, 500 students voted on each of two issues. Of these students, 375 voted in favour of the first issue, 275 voted in favour of the second, and 40 students voted against both issues. How many students voted in favour of both issues?

The following is a conversation between Esther and Frank who met on a bus.

- Esther: I have three children. Their ages are whole numbers, the product of their ages is 72, and the sum of their ages is the bus number of this bus that we are on.
- Frank: Of course I know the bus number, but I still don’t know their ages.
- Esther: Oh, I forgot to tell you that my two youngest children have the same age.
- Frank: Oh, now I know their ages.

So what are the ages of the three children?

**Editorial input note:** Enter the three ages in nondecreasing order, separated by commas, with no spaces.