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

Efficiency of pipe B is 60% more than that of A. If both the pipes are opened simultaneously, they take 20 minutes to fill the tank. How long will pipe A alone take to fill the same tank?

A sequence is created by beginning with and repeatedly turning the arrow $45^\circ$counter clockwise. The first terms of the sequence are
$$\to,\nearrow,\uparrow,\nwarrow,\leftarrow,\swarrow,\ldots$$
Determine the 2019th figure in the sequence.

In rectangle $ABCD$, $AD=1$, $P$ is on $AB$, and $DB$ and $DP$ trisect $\angle ADC$. What is the perimeter of $\triangle BDP$?

A $5\times7$ rectangle is partitioned into $1\times1$ squares. Then two opposite $1\times1$ corner squares are removed resulting in the figure below. How many squares with integer length sides can be found using the line segments in the resulting figure?

Father is aged three times more than his son, Sunny. After 8 years, he would be two and a half times of Sunny’s age. After further 8 years, how many times would he be of Sunny’s age?

Say that a whole number is “five-important” if it is a multiple of 5 and/or contains the digit 5. For instance, the numbers 55, 120, and 456 are five-important, but the number 2019 is not. How many different five-important numbers are there between 1 and 2019?

Samantha offers two different packages of yoga classes at her yoga studio. She offers two “hot yoga sessions” and three “zero gravity yoga sessions” at a total cost of \(\$400\). She also offers four hot yoga sessions and two zero gravity sessions at a price of \(\$440\). Samantha wants to offer a larger package for the long-time clients in which the cost must exceed \(\$800\). If Samantha does not wish to include more than 13 sessions for the long-time client package, will she be able to create this package for her clients?

Find the remainder when:

a) $1532^5-1$ is divided by 9.

b) $2016^{2018}+2$ is divided by 5.

c) $4^{2018}-7$ is divided by 9.

d) $41^{2015}-6$ is divided by 7.

**Editorial input format:** Enter the four remainders in a, b, c, d order, separated by commas.

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

A 9-by-12 rectangular picture is framed by a border of uniform width. Given that the combined area of picture plus frame is 208 square units, find the width of the border.

Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob, and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $1/6$, independent of the outcomes of any other toss.)

In a list of seven integers, 13 is the smallest member, 37 is the largest member, the mean is 23, the median is 24, and the mode is 18. If the numbers 8 and 43 are then added to the list, which of the following will change: the mean, the mode, or the median?

Tom writes three different two–digit numbers on the board. The three tens digits of the three numbers are 1, 2, and 3, and so are their three unit digits. However, none of the three numbers has the same tens and unit digit. One of the three numbers are 12. What are the other two numbers?

**Editorial input format:** Enter both other numbers, separated by a comma.

**Editorial input note:** Enter the smaller number first, with a comma and no spaces between the two numbers.

What is the largest product that can be formed from using the digits 2, 3, 4 and 5, and one multiplication sign? You are only allowed to combine the digits to form two numbers, e.g. $2\times345$, but you are not allowed to use indices, e.g. $2^3\times4^5$ is not allowed.

The numbers $a_1,a_2,a_3$ and $a_4$ are drawn one at a time from the set $\{0,1,2,\ldots,9\}$. If these four numbers are drawn with replacement, what is the probability that $a_1a_4-a_2a_3$ is an even number?

A student has to answer 10 questions, choosing at least 4 from each of Parts A and B. If there are 6 questions in Part A and 7 in Part B, in how many ways can the student choose 10 questions?