AMO 2027 Grades 8-9 SET 10 - Self-study

What is the remainder of $1234^{2017}:7$?

Susan jogged from home to the sport complex at a constant speed of 6 km/h. 10 minutes later, her sister, Susie, started riding a bicycle along the same route towards the sports complex at a constant speed of 18 km/h. If they arrived at sports complex at the same time, how many seconds does it take Susan to jog from home to the sport complex?

Find how many three-digit numbers satisfy all the following conditions:
- if it is divided by 2, the remainder is 1,
- if it is divided by 3, the remainder is 2,
- if it is divided by 4, the remainder is 3,
- if it is divided by 5, the remainder is 4,
- if it is divided by 8, the remainder is 7.

In the figure below, $BAD$ is a right angle, and $C$ is the midpoint of segment $AB$. Given the dimensions marked in the figure, the length of $CD=\sqrt a$. Find $a$.

Find the sum of the following series:
$$1+3+5+7+\ldots+197+199.$$

Two circles with equal radii are enclosed by a rectangle, as shown. The distance between their centres is $\frac{2x}{3}$. The value of $x$ is

The figure shows a rectangle $PRSU$ with a line $QT$ which divides the rectangle into two squares.
How many right triangles can be drawn using any three points $P,Q,R,S,T,U$ as corners?

In a pair of fair Wacky Dice, the red die has 1, 1, 1, 2, 3, and 3 on its six faces, while the green die has 2, 3, 3, 3, 5, and 5 on its six faces. If a pair of fair Wacky Dice is rolled, what is the probability the sum is odd?

The positive integers are arranged in rows and columns as shown below.



More rows continue to list the positive integers in order, with each new row containing one more integer than the previous row. How many integers less than 2000 are in the column that contains the number 2000?

Henry Carol and James have an average of \(\$260\). Altogether, Henry and Carol have \(\$80\) more than James. Carol has \(\$130\) more than Henry. How much does James have?

In a tank, there is a pipe which can be used for filling the tank as well as emptying it. The capacity of the tank is $6000\mathrm{m}^3$ (meter cube). The emptying capacity of the pipe is $10\mathrm{m}^3$ higher than its filling capacity and the pipe needs 20 minutes lesser to empty the tank than it needs to fill the tank. What is the filling capacity of the pipe?.

Among three students A, B, and C, one comes from America, one comes from Canada and one comes from Singapore. Also, one of them is in Grade 6, one is in Grade 7 and one is in Grade 8. It is known that A is not in Grade 6, C is in Grade 8, the Grade 7 student is from Canada, the Grade 6 student is not from America. Which grade is B in? Where is B from?

**Editorial input format:** Enter both the grade number and country, separated by a comma.

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

Find the missing number.

a) $10,30,32,96,98,294,(\quad),(\quad),\ldots$

b) $5,16,49,148,(\quad),(\quad),\ldots$

c) $6,10,7,11,8,12,(\quad),(\quad),\ldots$

d) $5,10,17,26,37,50,(\quad),(\quad),\ldots$

**Editorial input format:** Under the simple rules explained in the solution, enter two values for each part in a–d order. Use commas within pairs and semicolons between pairs.

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

In the figure, $AB=AC=AD$, $\angle ABC=70^\circ$, and $\angle ADC=50^\circ$. Find the measure of $\angle BAD$.

Randy and Sandy both have a total of 20 books. After Sandy loses three by leaving them on the bus, and some birthday gifts double Randy's collection, their total increases to 30 books. How many books did each have before these changes?

**Editorial input format:** Enter the original counts in Randy, Sandy order, separated by a comma.

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