AMO 2027 Grade 6 SET 1 - Self-study

The first 73 odd whole numbers are written down. How many times does the digit 3 appear? Count every occurrence, so a number with two 3s contributes twice.

**Input:** Choose one option.

One light flashes every 7 minutes and another light flashes every 2 minutes. If both lights flash together at 1 PM, how many minutes after 3 PM will both lights flash together for the first time?

**Input:** Enter the integer only, without units.

Find the sum of the digits of the next number in the sequence $11,143,2431,46189,\ldots$.

Use the intended pattern of cumulative products of successive prime numbers greater than 10.

**Input:** Enter the integer only, without units.

Look at the table. Which two cars are going at the same speed?

- Car A: 40 km in 1 hour
- Car B: 30 km in 2 hours
- Car C: 40 km in 2 hours
- Car D: 20 km in 30 minutes



**Input:** Choose one option.

How many 3-digit whole numbers have their digits in strictly decreasing order from left to right?

**Input:** Choose one option.

All six faces of the cube shown are painted black. It is then cut into 64 identical smaller cubes. How many smaller cubes have black paint on at least two faces?



**Input:** Choose one option.

The same symbol always has the same value. Given
$$\square+\triangle+\diamond=60,\quad \triangle+\diamond+\star=44,$$
$$\diamond+\star+\square=52,\quad\triangle+\diamond+\star+\square=74,$$
find $\star+\square+\triangle$.



**Input:** Choose one option.

The interest on 1200 dollars exceeds the interest on 1000 dollars by 30 dollars over 3 years. Find the annual interest rate. Both deposits use the same rate and interest model, but the question does not specify simple interest or compounding. For this question, do not assume an unstated model; decide whether the information determines a unique annual rate.

**Input:** Choose one option.

Ten standard dice are rolled and their top-face values are added. Each die has the six face values 1, 2, 3, 4, 5 and 6. How many different sums are possible?

**Input:** Choose one option.

A sports club offers soccer, basketball and baseball. Every member plays at least one sport. A member may play any one, any two or all three sports, with no restrictions on combinations. Which diagram can represent all these possible membership combinations? The area outside the circles contains no club members.



**Input:** Choose one option.

Ananya wants to tile a floor that is 24 m by 40 m. There are two types of tiles: a square that is 2 m on a side and an L-shape as shown. The L-shaped tile can be turned over or rotated if needed. What is the least number of tiles Ananya needs to tile the floor completely?

All indicated tile lengths are in meters. Tiles cover the floor without gaps or overlaps.



**Input:** Enter the integer only, without units.

Consider the figures below. How many dots will the 22nd figure contain?



Continue the illustrated diamond-growth rule: each new figure adds the next complete outer diamond ring of equally spaced dots.

**Input:** Choose one option.

Find the value of $\displaystyle\frac{\frac1{2+\frac12}+\frac1{1+\frac14}}{1-\frac25}$.

**Input:** Enter the integer only, without units.

Phineas forms an ordered list of seven consecutive whole numbers. The sum of the first, third and sixth numbers is 175. Find the sum of the remaining four numbers.

**Input:** Choose one option.

The numbers 123456789 and 987654321 are added. This sum is divided by a single-digit number, leaving no remainder. How many possible single-digit numbers could have been used?

Here “single-digit number” means a positive digit from 1 to 9.

**Input:** Choose one option.