AMO 2027 Grades 6-7 SET 5 - Self-study

The diagram shows a diagonal of a 4-by-3 rectangle passing through 6 unit squares. Find the number of unit squares passed through by a diagonal of a 18-by-12 rectangle.

Two towns, A and B, lie to the north of a straight road running in the East-West direction (as shown in the diagram below). Unfortunately they are neither connected to the road nor to each other. The people living in the two towns decide to build two new roads, one from each town, to the existing road, such that the route connecting the two towns via the existing road will be the shortest. Given that AC = 10 km, BD = 20 km and CD = 40 km, find the length of the shortest route from town A to town B.

Hannah used some coins to make a octagon. There were 119 coins on each side of the octagon. There were one coin at every corner. How many coins were used to make the octagon?

The figure below is made up of 28 identical squares. If the area of the figure is 1008cm², find the perimeter of the figure.

Which one of the following integers can be expressed as the sum of 100 consecutive positive integers?

A stamp collecting club calculated that the average number of stamps in its members’ 10 collections was 88. However, it was discovered that 2 numbers in the calculations were entered incorrectly. The number 55 was entered as 75 and the number 78 as 88. What is the correct average number of stamps in the 10 collections?

Mr. Lee left Town X in a van at 8.30 a.m and traveled at a constant speed of 50 km/h. Three hours later, Mr. Wee took the same route by car and traveled at a constant speed. After traveling 300 km, Mr. Wee caught up with Mr. Lee.

a) What is Mr. Wee’s speed?

b) How far apart would they be at 1 : 30 p.m?

Masses (kilograms)

| Person | Rock 1 | Rock 2 | Rock 3 | Rock 4 | Rock 5 | Rock 6 |
| --- | --- | --- | --- | --- | --- | --- |
| Andrew | 2.4 | 2.5 | 3.6 | 3.1 | 2.5 | 2.7 |
| Maria | x | 3.1 | 2.7 | 2.9 | 3.3 | 2.8 |

Andrew and Maria each collected six rocks, and the masses of the rocks were shown in the table above. The mean of the masses of the rocks Maria collected was 0.1 kilogram greater than the mean of the masses of the rocks Andrew collected. What was the value of x?

How many diagonals are there in a pentagon (5-sided polygon)?

Gina’s 6-digit postal code has the following interesting properties:

- It is the same number if your read it from right to left.

- It is a multiple of 9.

- If you remove the first and last digits, the only prime factor of the remaining 4-digit number is 11.

Find the first 3 digits of Gina’s postal code.

Given that $15ABC6$ is the smallest six-digit number that starts with “15”, ends with “6”, and is divisible by 36. Find the value of $A^2+B^2+C^2$.

Joe and JoAnn each bought 12 ounces of coffee in a 16 ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe’s coffee to that in JoAnn’s coffee?

In a basketball game, Marc, Jackson, and Cole scored 106 points altogether. Marc scored 15 more points than Jackson. Jackson scored 7 fewer than Cole. How many points did Cole score?

A carton of 17 similar packs of screws has a mass of 33.95kg. If the mass of the empty carton is 800g, find the mass of 6 such packs of screws.

The diagram shows a quadrilateral ABCD, in which $AD=BC$, $\angle CAD=50^\circ$, $\angle ACD=65^\circ$, $\angle ACB=70^\circ$. What is the size of $\angle ABC$?