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

There are 11 trees along the road in front of Ksi’s house. The trees are 10m away from each other. How far is the eleventh tree from the first one?

In the following alphabetic, all different letters stand for different digits. Find the sum of P + E + M + C + E.

$\overline{UPA}+\overline{UCAA}=\overline{PEMCE}$.

Mandy writes all counting numbers from 1 to 202 on the blackboard. How many times does the digit 2 appear on the blackboard?

Max cuts a string 8 times so that each resulting string has an equal length of 12cm. How long is the original string?

Christine thought of a number. She add $\frac{1}{3}$ of this number and then multiplied the result by 5. Then she subtracted 4 and then divided the result by 8. She got 32 in the end. What was the original number?

Pi and Ksi had 250 marbles altogether. After Pi had given $\frac{1}{6}$ of his marbles to Ksi, Ksi had the same number of marbles as Pi. How many marbles did each boy have at first?

Answer entry: Enter Pi; Ksi, in that order, separated by a semicolon.

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

The table shows the number of books read by 4 groups of children in a week.

| Group | Number of children | Number of books each read |
|---|---|---|
|1|26|1|
|2|15|2|
|3|8|3|
|4|5|4|

Which group of children read 20% of the books?

Town A and Town B were 880 km apart. A car travelled from Town A to Town B at a speed of 100 km/h. At the same time, another car travelled from Town B to Town A at a speed of 120 km/h. How long would it take for the two cars to be 110 km apart?

Answer entry: give a time in hours; decimals or fractions are allowed.

Pi sold 400 eggs on Sunday, and 15% more eggs on Monday than on Sunday. On Tuesday, Pi sold 5% more eggs than on Monday. How many eggs did he sell on the three days altogether?

The bar graph shows the favorite colors of 20 students in a class.
How many more of them favored orange than those who favored green?

The digits 2, 4, 6 and 8 are each used once to create two 2-digit numbers. What is the largest possible difference between the two 2-digit numbers?

Driving from Boston to New York one day, Sara covered the 250 miles in five hours. Because of heavy traffic, the 250-mile return took six hours and fifteen minutes. Calculate average speeds for the trip to New York, the trip from New York, and the round trip.

Answer entry: Enter outbound; return; round-trip speeds in mph, in that order, separated by semicolons. Use the exact fraction for the last answer, or round it to one or two decimal places.

**Editorial input note:** Use no spaces before or after semicolons. For a fractional final answer, use a fraction in lowest terms. Omit unit labels.

A rectangle is divided into four smaller rectangles with areas as shown in the diagram below. Find the value of A.

Evaluate

a) $14 + 28 + 42 + 56 + \cdots + 700$

b) $1 + 2 − 3 + 4 + 5 − 6 + 7 + 8 − 9 + \cdots + 28 + 29 − 30$

c) $(2 + 4 + 6 + \cdots + 28 + 30) − (1 + 3 + 5 + 7 + \cdots + 27 + 29)$

d) $102 − 105 + 108 − 111 + \cdots + 144 − 147 + 150$

e) $1 \times 2 \times 3 + 2 \times 4 \times 6 + 3 \times 6 \times 9 + \cdots + 10 \times 20 \times 30$

f) $9751 \times 419 − 9700 \times 400 − 9751 \times 19$

Answer entry: Enter the six answers (a) through (f), separated by semicolons, without thousands separators.

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

The school nurse is recording the heights of students in a class. When comparing their heights, the students find that:

Alex is taller than Brian but shorter than Charlie.

Daniel is taller than Edward but shorter than Alex.

Who is the tallest among 5 students? In your answer sheet, write 001 for Alex, 002 for Brian, 003 for Charlie, 004 for Daniel and 005 for Edward.