AMO 2027 Grades 4-5 SET 18 - Self-study

Evaluate $T$:

$T=(7+11+15+\cdots+1831+1835)$

$\qquad-(14+18+\cdots+1822+1826)$.

Which of the following options shows the correct number of minutes in a 1-day period?

Use a standard 24-hour day.

In the figure below, the shaded part is the overlapping part of the two identical squares. The perimeter of the shaded part is 26cm. What is the perimeter of the rectangle?

The bar graph below shows the number of burgers sold on Monday. Study it carefully and answer the questions that follow:

a) Which burgers were sold more than 100 but less than 180?

b) What is the total number of the 3 most popular burgers sold?



Answer entry: Use FF for Fish Fillet, C for Cheese, DC for Double Cheese, SC for Samurai Chicken and SB for Super Burger. Enter the codes for part(a), comma-separated in graph order, then a semicolon and the integer total for part(b). No spaces.

Town A and Town B are 378km apart. At 9 : 45 a.m., Roy set off from Town B to Town A at an average speed of 88km/h. 15 minutes later, Sam set off from Town A to Town B at an average speed of 90km/h.

a) How far from Town A was Roy at 10a.m?

b) At what time did Sam and Roy pass at each other?

Motion convention: both people keep the given constant speeds along the same road, without stops.

Answer entry: Enter a;b, with a in kilometres and b as a 24-hour HH:MM time; no units or spaces.

A road is 120m long. If trees are planted on both its sides at intervals of 5m. How many trees are planted on the both sides of the road if both ends of the road are also planted with trees?

Tom is 6 years older than Nina. Five years from now, the sum of their ages will be 52. How old are both of them?

Find their ages now; retain the statement’s “five years from now” condition.

Answer entry: Enter Tom;Nina as two integer ages in years, separated by a semicolon, without spaces.

Susie had some red and blue marbles. 80% of the marbles were red. After she bought another 63 red marbles and 46 blue marbles, 75% of the marbles were red. Find the total number of marbles she had at first.

Naomi, Olivia, Pam and Rina were from the same class. One of them was the last person who left the class and did not turn off the fan. When the teacher checked with them the next day, their replies were as follows:

• Naomi: Pam was still in the classroom when I left.

• Olivia: Rina wasn’t there when I left the classroom.

• Pam: Olivia had already left when I left the room.

• Rina: I didn’t really pay much attention.

Who was the last person who left the classroom if all of them told the truth?

Departure convention: each person was in the classroom and left it once, without returning.

If the four-digit number $\overline{19N6}$ is divisible by 14, find N.

N is a base-ten digit, including 0 as a possible digit to test.

Two pipes can fill a tank in 10 hours and 12 hours, respectively while a third pipe empties the full tank in 20 hours. If all the three pipes operate together, in how much time the tank will be filled, in minutes?

Tank convention: the tank is initially empty and all three pipes work at their stated constant rates throughout.

According to the graph below, in which year was the ratio of the number of students enrolled at School B to the number of students enrolled at School A the greatest?

There are 14 rows of seats in a school hall. There are 70 seats in the last row. If there is an increase of 3 seats, starting from the first row, how many seats are there in the school hall?

Row convention: each successive row has exactly 3 more seats than the row before it.

Billy and Phil the rabbits are in separate cages. The ratio of the number of carrots in Billy’s cage to Phil’s cage was 3 : 4. After some carrots from Phil’s cage were transferred to Billy’s cage, the ratio of the number of carrots in Billy’s cage to Phil’s cage became 2 : 1. What is the smallest possible number of carrots that were transferred?

Count whole carrots; carrots are transferred without splitting and none are added or removed.

In the following cryptarithm, all different letters stand for different digits. Find the sum (A + B + C).

$\overline{6A}+\overline{BC}=\overline{A09}$.

Each letter is one base-ten digit; A and B, as leading digits, are nonzero. Distinct letters have distinct digits.