AMO 2027 Grades 2-3 SET 13 - Self-study

How many triangles are there in the following figure?

Compute the following sums:

a) 1 + 2 + 3 + 4 + 5.

b) 1 + 2 + 3 + 4 + ⋯ + 18 + 19 + 20.

c) 2 + 5 + 8 + 11 + ⋯ + 32 + 35 + 38.

Self-study input convention (editorial): enter all three answers in order a;b;c, separated by semicolons.

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

Early in the morning, the hen went out to buy rice. When she returned home, she realized that all her eggs were missing. She approached the smart monkey for help. The monkey then identified 3 suspects: the fox, the wolf and the mouse.

• Fox: “I didn’t steal the eggs.”

• Mouse: “It is the fox who stole it.”

• Wolf: “The mouse is lying.”

If only one of them was telling the truth, can you help the monkey identify the one who stole the eggs?

Double bar graph show the number of students in MN high school who did worked part time while studying. Use the graph to solve the problems given below the graph.



a) What is the difference between the number of working and non-working students in the group of 15?

b) Which age group worked the most (in terms of number of students)?

Self-study clarification and conditional input (editorial): the original chart has no legend identifying working students. Answer part (a), then give part (b) for each possible series. Enter three numbers separated by semicolons: difference at age 15; age with the largest left bar; age with the largest right bar. This input preserves both conditional interpretations and does not assign a missing legend.

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

The stretch of road leading to Pi’s house is to be planted with some trees at regular intervals of 15m, and trees are planted at opposite ends of the road. If the road is 900m long, how many trees are to be planted along that stretch of road?

Self-study interpretation (editorial): count one row of trees, including both endpoints of the road.

The directions on an exam allow 2 hours and 30 minutes to answer 50 questions. If you want to spend an equal amount of on each of the 50 questions, about how much should you allow for each one?

Self-study input (editorial): enter the time per question in minutes.

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



Find the 4-digit number GOOD.

The sum of the ages of 5 children born at the intervals of 3 years each is 50 years. What is the age of the youngest child?

Alice got on the school bus. At the stop after Sarah's, 7 students got on. Five students got on the bus at the next stop. At the last stop before the school, 9 students got on. When the bus arrived at school, 38 students got off. How many students were already on the bus when Sarah got on?

Self-study assumptions (editorial): treat Alice and Sarah as the same child; assume that only this child boards at that stop, there are no other unmentioned boarding or leaving events, and all students leave at school. Enter the number already aboard before that child boards.

The figure below is made up of 28 identical squares. If the area of the figure is 1008cm², find the perimeter of the figure.



Self-study input (editorial): enter the perimeter in centimeters.

Find the smallest 2 - digit number divided by 3 and 5 with a remainder of 2.

Pi used black and white triangles to form some patterns. The first four patterns are shown below.



Find the total number of unshaded triangles for pattern 10.

Self-study growth rule (editorial): continue the displayed construction, adding successive rows of 1, 2, 3, and so on unshaded small triangles.

Amy, Bob and Chris each belong to one of the two tribes: truthtellers, people who always tell the truth and liars, people who always lie. Amy and Bob made the following the statements:

Amy: “All of us are liars.”

Bob: “Exactly one of us is truthteller.”

Which tribe does each of them belong to?

252 children took part in a mathematics competition. 65 of them are girls. How many more boys than girls were there?

Each of the numbers 3, 4, 5, 6, 7 and 8 is to be placed in each circle in the below diagram. The sum of the numbers in each square is 20. What is the sum of the numbers in the shaded circles?



Self-study clarification (editorial): use each of the six numbers exactly once.