Question 1
The figure below is divided into 8 congruent squares as shown. The area of the figure is 72cm². What is the length of the darkened border in centimeters?
AMO 2027 Grades 2-3 SET 12 - Self-study
Question 2
Alice had several essays to write this weekend. Her father asked how many essays she needed to write, and she made two statements:
• Statement 1: “I had at least 3 essays to write.”
• Statement 2: “I had at most 3 essays to write.”
If her father knew that Stella never lies to him, could he determine exactly how many essays Stella needed to write on this weekend?
Self-study correction and input (editorial): assume Alice and Stella are the same child and both statements are truthful. Enter the exact number of essays.
• Statement 1: “I had at least 3 essays to write.”
• Statement 2: “I had at most 3 essays to write.”
If her father knew that Stella never lies to him, could he determine exactly how many essays Stella needed to write on this weekend?
Self-study correction and input (editorial): assume Alice and Stella are the same child and both statements are truthful. Enter the exact number of essays.
Question 3
How many triangles are there in the following diagram?
Question 4
What is the next number in the sequence below?
20, 10, 16, 7, 12, 4, 8, ...
Self-study rule (editorial): terms in odd positions decrease by 4; terms in even positions decrease by 3.
20, 10, 16, 7, 12, 4, 8, ...
Self-study rule (editorial): terms in odd positions decrease by 4; terms in even positions decrease by 3.
Question 5
Pi has twice as much money as Matthew. Pi has four times as much money as Sally. Sally has \(\text{\$3}\) more than Andrew. If Matthew has \(\text{\$14}\), how much money do Andrew, Sally and Pi have?
Self-study input convention (editorial): find all three individual amounts, then enter their combined total in dollars. The solution gives both the individual amounts and the total.
Self-study input convention (editorial): find all three individual amounts, then enter their combined total in dollars. The solution gives both the individual amounts and the total.
Question 6
Look at each pattern of shapes and see if you can predict the following:
What will shape number 35 look like?
Self-study rule (editorial): repeat the visible four-shape cycle. Enter the name of shape 35.
What will shape number 35 look like?
Self-study rule (editorial): repeat the visible four-shape cycle. Enter the name of shape 35.
Question 7
Mrs. Walker and Ksi the student make cookies for the school bake sale. Mrs. Walker baked 86 cookies. Ksi baked twice as many as Mrs. Walker. How many cookies did they bake altogether?
Question 8
a) Books are on sale for \(\text{\$7}\). Peter has \(\text{\$30}\) in his wallet. How many books can he buy?
b) A resting dolphin needs to take a breath 3 times a minute. How many minutes would it take to make 40 breaths?
c) A snail has fallen in a 10 feet deep hole. It started climbing up - every day it would climb 3 feet up but then it would fall 2 feet down while sleeping during the night. After how many days will the snail climb out of the hole?
Self-study timing and input convention (editorial): for part (b), assume evenly spaced breaths, with the first counted breath after 20 seconds. Enter three numbers separated by semicolons in order a;b;c: books; elapsed seconds for (b); days. The solution also expresses (b) in the minutes requested originally.
**Editorial input note:** Use no spaces before or after semicolons.
b) A resting dolphin needs to take a breath 3 times a minute. How many minutes would it take to make 40 breaths?
c) A snail has fallen in a 10 feet deep hole. It started climbing up - every day it would climb 3 feet up but then it would fall 2 feet down while sleeping during the night. After how many days will the snail climb out of the hole?
Self-study timing and input convention (editorial): for part (b), assume evenly spaced breaths, with the first counted breath after 20 seconds. Enter three numbers separated by semicolons in order a;b;c: books; elapsed seconds for (b); days. The solution also expresses (b) in the minutes requested originally.
**Editorial input note:** Use no spaces before or after semicolons.
Question 9
The sum of the digits of a three-digit number is 6. The hundreds digit is twice the units digit, and the tens digit equals the sum of the other two. Find the number.
Question 10
On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working 20 days?
Question 11
Ten lampposts are equally spaced along a straight line. The distance between two adjacent lampposts is 20 m. What is the distance between the first and the last lamppost?
Question 12
The following table shows the scores of Tiffany, Ursula, Vincent and Warren in the “Mathcraft Game”.
Who received the highest score?
Self-study interpretation (editorial): compare each player’s total score across all four rounds.
Who received the highest score?
Self-study interpretation (editorial): compare each player’s total score across all four rounds.
Question 13
A movie is scheduled for two hours. The theater advertisements are 4 minutes long. There are two previews; one is 5 minutes long and the other is 3 minutes. The rest of is devoted to the feature. How long (in minutes) is the feature film?
Question 14
Pi goes for a morning walk every day. He walks along a path whose total length is 56 ft. Find the length of the missing side.
Question 15
Gary adds two 2-digit numbers. In one of the numbers, he accidentally reads digit 2 as 5 in the ones place and the digit 6 as 9 in the tens place. He gets 111 as the result. If Gary reads the digits correctly, what is the correct sum?