AMO 2027 Grades 4-5 SET 16 - Self-study

Josh lives on the 6th floor. He needs to climb 48 steps to move from the 2nd floor to the 4th floor. How many steps does he need to climb from the 1st floor to the 6th floor?

Usual staircase convention: each interval between adjacent floors has the same number of steps.

Two 9cm × 13cm rectangles overlap as shown to form a 9cm × 22cm rectangle. What is the area of the overlapping rectangular region?

The ratio of Don's money to Ahmad’s money is 4: 7. The ratio of Don’s money to Sally's money is 2 : 1. If they have 312 dollars altogether, how much money does Don have?

Evaluate the following expressions.

(a) $74\times15-15\times\frac{55}{4}+44\times15-15\times\frac{17}{4}$.

(b) $\frac{2+4+6+8+\cdots+200}{3+6+9+12+\cdots+300}$.

(c) $\frac1{1\times2}+\frac1{2\times3}+\frac1{3\times4}+\cdots+\frac1{2020\times2021}$.

(d) $\frac16+\frac1{12}+\frac1{20}+\frac1{30}+\cdots+\frac1{2070}$.

(e) $\frac1{1\times3}+\frac1{3\times5}+\frac1{5\times7}+\cdots+\frac1{2019\times2021}$.

(f) $1+\frac1{1+2}+\frac1{1+2+3}+\frac1{1+2+3+4}+\cdots+\frac1{1+2+3+\cdots+2020}$.

Answer entry: six simplified values in a;b;c;d;e;f order, using a/b for fractions and semicolons between values.

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

If the four-digit number $\overline{4N79}$ is divisible by 21, find N.

Zachary, Nicholas and Philip are old classmates. During a recent gathering, they told each other about their occupations. Among them, there is an accountant, a bus driver and a manager. Below are the hints about their occupations.

• Philip is older than the accountant.

• Zachary’s age is not the same as the bus driver’s age.

• The bus driver is younger than Nicholas.

Can you find out Nicholas’ occupations?

Rose saved 25 dollars in the first month, 40 dollars in the second month, 55 dollars in the third, and so on. The amount of money she saved in the last month was 190 dollars. How much money did Rose saved in total?

$2^5$ means 2 multiplied by itself 5 times, i.e. $2^5=2\times2\times2\times2\times2=32$. What is $3^4$ equal to?

Find the smallest whole number that is divisible by 5, and leaves a remainder of 1 when divided by 2 or by 3.

In Planet Trek, Chok is a 3-legged animal and Rogku is a 5-legged animal. If the total number of Chok and Rogku is 30 and their total legs is 128, how many Rogkus are there on Planet Trek?

$\frac13$ of the pupils in a hall were boys. When 35 boys left the auditorium, the number of boys decreased to $\frac14$ of the total number of pupils in the hall. How many girls were in the hall?

Find the area of the region.

A boat running downstream covers a distance of 16km in 2 hours while for covering the same distance upstream, it takes 4 hours. What is the speed of the boat in still water?

There are chickens and rabbits in a cage. Look at the top of the cage - there are totally 35 heads. Look at the bottom of the cage - there are 94 legs. How many rabbits are there in the cage?

Josie wrote a number sequence 2, 14, 26, 39, 50, 62 and said that the rule was adding 12. Her brother said, “Recheck the terms, one of them is out of the rule.” Can you explain why her brother said that?