AMO 2027 Grades 4-5 SET 8 - Self-study

Each figure is formed by surrounding one row of black squares with white squares. How many white squares are needed to surround one row of 50 black squares?

A typist use 181 digits for book’s page numbers. How many pages did the book have?

Find the sum of the terms in the 25th pair of brackets:

[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12], ...

A number sequence starts with 10 as the 1st term. The next term is obtained by first multiplying the current term by 2 and then subtracting 5 from the result. For example, the first four terms of the sequence are 10, 15, 25, 45, ... Find the difference between the 11th and the 12th term.

William puts some red, blue and white marbles into a box. The number of red marbles is 1/3 of the number of blue marbles. The number of white marbles is 1/4 of the number of red marbles. If there are 144 more blue marbles than red marbles, how many marbles are there in the box?

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?

In the following sequence, the difference between the consecutive terms is a constant. Find the value of c.

777, a, b, c, d, 832, ...

Find the area of the figure below which is made up of 7 identical rectangles of length 30cm.

16% of the balls in a basket are purple. 8/21 rest are yellow and the balance green. There are 48 more yellow balls than purple ones. How many balls are there in the basket altogether?

Find the missing number.

$\begin{array}{rrrr}5&2&3&10\\6&4&1&11\\1&9&?&12\\12&?&6&?\end{array}$



Answer entry: enter the three missing values in reading order (row 3 column 3; row 4 column 2; row 4 column 4), separated by semicolons.

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

The speed of sound in air is 1100 feet per second. The speed of sound in steel is 16500 feet per second. Robin, one ear pressed against the railroad track, hears a sound through the rail six seconds before hearing the same sound through the air. To the nearest foot, how far away is the source of that sound?

There are 3 boxes, exactly one of which has a car. You can keep the car if you pick the correct box!

On each box there is a statement, exactly one of which is true.

Box 1: The car is in this box.

Box 2: The car is not in this box.

Box 3: The car is not in box 1.

Which box has the car?

Find the digits represented by $a$ and $b$ in $\overline{a7889b}$ so that it is divisible by 15.

Answer entry: enter any valid pair a;b. The solution lists all valid pairs.

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

Evaluate $1-3+5-7+9-11+13-\cdots+97-99+101$.

The pictures below are made up of squares and 6-sided shapes, each of which is added one shape at a time. Figure 1 has 17 lines, figure 2 has 20 lines and figure 3 has 25 lines. How many lines will there be in figure 6?