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

A product of a number and itself is called a square number. Each pattern below is formed by identical squares. Which pattern will that be there are 100 squares?

What is the next number below?

3, 5, 6, 10, 9, 15, 12, 20, ...

Find the sum of the digits of the sum below.

4123 + 1234 + 2134 + 3214.

An octopus has 8 tentacles and 1 head. A jellyfish has 20 tentacles and no head. A cow has 4 legs and 1 head. Farmer Brown, who raises only octopus, jellyfish and cows on her farm, has animals with a total of 17 heads, 196 tentacles and 20 legs. How many animals does she have?

Sue was given some allowance. She spent half of it on food, $\frac{1}{3}$ of the remainder on a file and saved the rest. How much was her allowance if she was left with 5 dollars?

Mrs. Walker read 5 pages of storybook on the first day. She read 10 pages of the same storybook on the second day, 15 pages on the third day and so on. How many pages had the storybook at most if Mrs.Walker finished reading it on the tenth day?

The sum of two numbers is 340. Its quotient and remainder are 7 and 20 respectively when one of them is divided by the other. Find the two numbers.

Answer entry: Enter larger; smaller, separated by a semicolon.

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

A car went a distance of 90 km at a steady speed and returned along the same route at half that speed. The time needed for the whole round trip was four hours and a half. Find the two speeds.

Answer entry: Enter outbound; return speeds in km/h, separated by a semicolon.

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

The second day of a month is Sunday. What will be the last day of the next month which has 31 days?

Answer entry: state if the weekday cannot be determined, or give any weekday that can satisfy the information.

Five teams take part in a zonal basketball league. Each team has to play exactly one match against another team. So far, team A has played 4 matches and team B has played 3 matches. If team D and team E have each played 2 matches, how many matches has team C played?

Answer entry: give any possible number of matches for team C.

The original price of a watch at a shop was 80 dollars. After selling the watch at a discount of 25%, the shop owner still made a profit of 25%. What was the cost price of a watch?

For a certain 3-digit number:

• The digits are in increasing order.

• The difference of the greatest and least digit is 7.

• It is a multiple of 9 and greater than 200.

Find this 3-digit number.

In the following, all different letters stand for different digits.

$\overline{EYE}+Y=\overline{ODD}$.

Find the value of the 3-digit number ODD.

Find the sum of all odd numbers between 0 and 20.

A 200-meter long street has a tree every 4 meters on one side. How many trees are there if both ends of the road are also planted with trees?