AMO 2027 Grades 4-5 SET 9 - Self-study

Find the missing number.

a) 10, 30, 32, 96, 98, 294, ( ), ( ), …

b) 5, 16, 49, 148, ( ), ( ), …

c) 6, 10, 7, 11, 8, 12, ( ), ( ), …

d) 5, 10, 17, 26, 37, 50, ( ), ( ), …

e) 50, 49, 47, 44, 40, 35, ( ), …

f) 1, 2, 5, 10, 17, 26, ( ), …?

Answer entry: separate parts (a)–(f) with semicolons and numbers within a part with commas; enter ten numbers in total.

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

There are 25 trees planted at equal distances along a straight road. The distance between the first and the eighth trees is 56 meters. What is the distance, in meters, between the first and the last trees?

On a particular afternoon, Eugene and Benjamin left their school at the same time and headed towards the bus stop. After running at an average speed of 8m/s for 30 second, Eugene ran past their classmate Dexter. Dexter was walking in the opposite direction the bus stop back to school, as he had left his pencil case in school. 20 seconds later, Benjamin walked past Dexter. Given that Benjamin was walking at an average speed of 1m/s, find Dexter's average speed in m/s.

Pipes A and B can fill a tank in 5 and 6 hours respectively. Pipe C can empty it in 12 hours. If all the three pipes are opened together, then how many hours are there needed for the tank to be fulfilled?

Henry, Charlie and Daisy were classmates. One of them cleaned up the classroom one day while there was nobody around. Below were their replies when the teacher questioned about who cleaned up the classroom.

• Henry: ”Charlie cleaned the classroom.”

• Charlie: ”I did not do it.”

• Daisy: ”Neither did I.”

If one of them was telling the truth, find out who cleaned up the classroom.

The table shows the results of a research study that investigated the therapeutic value of vitamin C in preventing colds. A random sample of 300 adults received either a vitamin C pill or a sugar pill each day during a 2-week period, and the adults reported whether they contracted a cold during that time period. What proportion of adults who received a sugar pill reported contracting a cold?

Number of Adults Contracting Colds

| Group | Cold | No cold | Total |
|---|---:|---:|---:|
| Vitamin C | 21 | 129 | 150 |
| Sugar pill | 33 | 117 | 150 |
| Total | 54 | 246 | 300 |

In the following alphabetic, all different letters stand for different digits. Find $P$.

$I+I+I=\overline{PI}$.

There are 220 balls in a basket. 70% of them are red and half of the remainder are blue. What percentage of the balls is blue?

Answer entry: enter the numerical percentage, without the percent sign.

A number has a remainder of 2 when it is divided by 3. It has a remainder of 3 when it is divided by 4. Find the sum of two such numbers that are smaller than 30?

Given a sequence:

2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, …

a) What is the 2021st number?

b) What is the sum of the first 51 terms?

Answer entry: enter (a);(b), separated by a semicolon.

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

In a straight row, 12 trees are planted at equal distance. The distance between 1st and 12th trees is 33 meters. What is the distance between 3rd and 9th trees?

A book has 215 pages. How many digits are used to number all pages in the book, starting from 1?

A short street where Bernard lives has 6 houses. One day 35 letters are delivered in his street and he receives more letters than anyone else. What is the smallest number of letters he may have received?

What percentage of the whole triangle is shaded?

$\frac12$ of John's money is 45 dollars more than $\frac13$ of Kim's money. They have a total of 800 dollars. How much money does each of them have?

Answer entry: enter John;Kim, in dollars.

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