AMO 2027 Grades 8-9 SET 11 - Self-study

The diagram shows three touching circles each of radius 5cm and a line which touches two of them. Given that the total length of the perimeter of the two shaded shapes is $k\pi+t$. Find $k+t$.

The 5-digit number $\overline{2018U}$ is divisible by 9. What is the remainder when this number is divided by 8?

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were 71, 76, 80, 82, and 91. What was the last score Mrs. Walter entered?

Several Preps were meeting in a room. After 45 of them left, the room was $\frac58$ as full as it was initially. How many Preps were in the room at the start of the meeting?

At Euler Middle School, 198 students voted on two issues in a school referendum with the following results: 149 voted in favor of the first issue and 119 voted in favor of the second issue. If there were exactly 29 students who voted against both issues, how many students voted in favor of both issues?

Two pipes B and F can fill the tank in 10 hours and 15 hours respectively. If both the pipes are opened along with an outlet pipe M, then the tank is filled in 30 hours. How much time will the outlet pipe take to empty the full tank?

The number of ordered pairs $(a,b)$ of integers which satisfy the equation $a^b=64$ is

If the number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 43, what is the probability that the number will be divisible by 11?

The diagram below shows square $ABCD$ such that
$$AE=EF=FB=BG=GH=HC=CI=IJ=JD=DK=KL=LA=20\mathrm{cm}.$$
Find the area of the square $MNPQ$ in $\mathrm{cm}^2$.

Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn’t know her house number. She tells him, “My house number has two digits, and exactly three of the following four statements about it are true.”

i. It is prime.

ii. It is even.

iii. It is divisible by 7.

iv. One of its digits is 9.

This information allows Malcolm to determine Isabella’s house number. What is its units digit?

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

The average age of 5 boys is 13 years 6 months. A 16-year-old boy joins them. Find the average age of the 6 boys.

**Editorial input format:** Enter the average age in months (use 12 months per year).

Pipe A alone can fill a tank in 25 hours and pipe B alone can fill the same tank in 20 hours. At the beginning, both the pipes are opened simultaneously. After how much time A should be closed so that the tank is full in 12 hours?

The diagram shows parts of a city where the lines are roads. Charles walks from point $X$ to point $Y$ along the roads without ‘backtracking’ (i.e. he can only walk downwards or to the right in the diagram below). How many ways can he walk from $X$ to $Y$?

Find the perimeter and area of the figure below which made up of different circles.

**Editorial input format:** Enter both exact answers in perimeter, area order, separated by a comma. Use pi or π for the circle constant; omit units. Do not replace pi by an approximation.



**Editorial input note:** Use no spaces before or after commas.