AMO 2020 – Primary 5 (Grade 5)

Mr Bear had 257 dollars in the bank. He borrowed enough money to buy something for 249 dollars. That week he earned 529 dollars. He paid off his debt and put the rest in the bank. How many dollars does he now have in the bank?

Find $47\times12+12\times53-53\times8-8\times47$.

Five students guessed the number of marbles in a jar: Andrea 98, Bernice 105, Cathy 109, Darius 113 and Edward 115. Two guesses were exactly 3 away from the actual number. Who made the closest guess? Enter 001 for Andrea, 002 for Bernice, 003 for Cathy, 004 for Darius or 005 for Edward.

How many rectangles of any size, using the existing grid lines in the figure, contain an odd sum of the numbers inside them? A one-cell rectangle is included.

Find the hundreds digit of the sum of all positive multiples of 4 less than 100.

Find the first three digits of the greatest four-digit number whose digits strictly decrease from left to right, whose digit sum is 18, and which is a multiple of 15. For example, for 5342 you would enter 534.

Bon, Ton and Hon have three different two-digit prime numbers. Bon has the least and Hon the greatest. Each number has digit sum 10. What is Ton’s number?

In the sequence 5, 9, 13, 17, …, each term is 4 more than the previous term. What is the first three-digit term whose digits strictly increase from left to right?

In the cryptarithm shown, different letters represent different digits from 1 to 9. If T = 5, find the sum of the digits of the four-digit number PETS.

Find the first three digits of $576\times17+582\times15+1424\times17+1418\times15$.

A family has four children. Aidan’s age is the sum of Billy’s and Cheryl’s ages. Four years ago, Dorris’s age was the sum of Billy’s and Cheryl’s ages then. Eight years ago, Aidan was twice as old as Dorris then. Who is the oldest? Enter 001 for Aidan, 002 for Billy, 003 for Cheryl or 004 for Dorris.

Write the numbers 98, 99, 100, 101, …, 198, 199 consecutively as one long number, then delete every odd digit. How many digits remain?

Sally jogs 4 miles from home along a straight road in one hour. She then walks at a constant speed in the same direction. After one hour of walking she is 6.5 miles from home. How far from home, in miles, will she be after three more hours of walking?

Tom lists four consecutive multiples of a number. The average of the first two is 119 and the average of the last two is 187. What is the greatest multiple in his list?

The decimal expansion of $5/13$ is $0.384615384615\ldots$, with the block 384615 repeating. What is the 2020th digit after the decimal point?

At a diner, lunches have one fixed price and dinners another. Four lunches and three dinners cost 58 dollars. The next line in the source reads “and lunches and 4 dinners cost 75 dollars”, with the number of lunches missing.

For the intended version consistent with the answer key, use five lunches and four dinners costing 75 dollars. Find the price of one lunch in dollars.

Identical small squares surround a larger square as shown. Find the total number of rectangles formed by the drawn lines.

What is the second smallest three-digit whole number that is a multiple of 3, one less than a multiple of 4, and one more than a multiple of 5?

How many three-digit numbers are one more than a multiple of 9 and one less than a multiple of 10?

A rectangular prism has volume 240 cubic centimetres. A smaller rectangular prism is removed. Each of its three edge lengths is half the corresponding edge length of the larger prism. Find the remaining volume in cubic centimetres.