AMO 2027 Grades 8-9 SET 1 - Self-study

The number 1000 can be written as the product of two positive integers, neither of which contains zeros. The sum of these two integers is

Geoff is trying to unlock the forgotten passcode for his luggage. The passcode consists of 3 digits, each of which is from 0 to 9.
- He tried 923, only one digit is correct and correctly placed.
- He tried 958, only one digit is correct but wrongly placed.
- He tried 379, only two digits are correct but wrongly placed.
- He tried 142, none of the digits is correct.
- He tried 127, only one digit is correct but wrongly placed. What is the passcode?

How many integers between 1000 and 9999 have four distinct digits?

A list of whole numbers from 1 to 2015 is written on a sheet of paper. All the multiples of 5 are then struck off from the list. What is the last digit of the product of the remaining numbers?

The average mass of Mona, Amanda, Jack and Brian is 68kg. The mass of Mona and Amanda is 36kg less than the mass of Jack and Brian. If Mona’s mass is 56kg, find the mass of Amanda.

Luke has played 20 games and has a 95% winning percentage. Without losing any more games, how many more games in a row must he win to reach exactly a 96% winning percentage?

In the triangle shown, the first diagonal line, 1, 2, 3, 4, ..., begins at 1 and each number after the first is one larger than the previous number. The second diagonal line, 2, 4, 6, 8, ... begins at 2 and each number after the first is two larger than the previous number. The nth diagonal line begins at n and each number after the first is n larger than the previous number. In which horizontal row does the number 2025 first appear?

In the figure below, it is given that $\angle ABC=70^\circ$ and $\angle DAF=\angle DFA=\angle ECG=\angle EGC$. Find the sum of the four shaded angles.

A piece of string is cut in two at a point selected at random. What is the probability that the longer piece is at least $x$ times as large as the shorter piece ($x\ge1$)?

In the diagram, a square is split into four smaller regions: $A,B,C$ and $D$. The shaded regions $B$ and $D$ are squares and the difference between their areas is $133\text{ cm}^2$. If the sum of the perimeters of regions $A,B,C$ and $D$ is 152 cm, find the sum (in $\text{cm}^2$) of the areas of the shaded regions $B$ and $D$.

Some students want to order shirts with their school logo. One company charges 9.65 dollars per shirt plus a setup fee of 43 dollars. Another company charges 8.40 dollars per shirt plus a 58-dollar fee. For what number of shirts would the cost be the same?

Tap A is thrice as fast as tap B. If both the pipes are opened together, the time taken to fill the cistern is 30 hours. Find the time taken to fill the tank by A alone?

The sides of Fran’s square are 5cm longer than the sides of Tate’s square. Fran’s square has 225 sq cm more area. What is the area of Tate’s square?

In the game “Golden Die”, you are given two turns of tossing a cube die. If you get 1 or 6 in the first toss, you win. As a result, you don’t need to use the second turn. Otherwise, you have to toss the die once more. You win if and only if the result of the two tosses are the same. What is the probability that you win the game?

The median of the list $n,n+3,n+4,n+5,n+6,n+8,n+10,n+12,n+15$ is 15. What is the mean of the list?