AMO 2027 Grades 6-7 SET 23 - Self-study

Find the possible value(s) of $\sqrt{\sqrt{0.0081}+\sqrt{0.0001}+\sqrt{0.81}}$.

**Input:** Choose one option.

How many two-digit prime numbers have only odd numbers as their digits?

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

Compute the value of $\dfrac{2019^2-2000^2-19^2}{2000\times19}$.

**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.

Remi had a square garden with an area of 16 square meters. She extends the length of the garden by 3 meters and the width by 2 meters. What is the new perimeter, in meters, of the garden?

**Input:** Choose one option.

A paper is folded as shown below and punched in its folded state. When the paper is opened out again, what will be the pattern of holes formed?

The original fold sequence and the four original pattern options are shown below.



**Input:** Choose one option.

The absolute value of $N$, symbolized by $|N|$, represents the distance from the signed number $N$ to the origin, without regard to sign. For example, $|+5|=5$ and $|-3|=3$. Find the sum of the integer values of $N$ that satisfy $2<|N-3|<5$.

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

A steel cuboid is reshaped into a cube. Initially, its length, breadth, and depth were 270 cm, 100 cm, and 64 cm respectively. Find the sum of digits of the surface area of the cube.

Assume no steel is lost or added and neither shape contains voids. Use the numerical surface area in square centimeters.

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

What is the value of $80\times3-70\times4+60\times5-50\times6+40\times7-30\times8+20\times9$?

**Input:** Choose one option.

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.

A football league has 10 teams. During the season, each of the 10 teams plays exactly 3 games with each of the other teams. What is the total number of games played?

**Input:** Choose one option.

Evaluate $(5\times7\times2\times17)\div(14\times34\times35)$.

Original printed options: A. $14$; B. $0.71$; C. $0.071$; D. $17$; E. $\dfrac1{17}$.

**Input:** For this self-study version, enter the exact value as a fraction a/b, without units, rather than an option letter or a rounded decimal. The original options are retained above for comparison.

I have a set of four numbers $P,Q,R,S$ in ascending order. The average of $P,Q,R$ is 22, and the average of $Q,R,S$ is 24. The largest number is 27. Then the smallest number in this set:

**Input:** Choose one option.

In the following cryptarithm, each different letter represents a different digit. Assuming a number cannot start with 0, find the last three digits of the smallest possible sum. For example, if your answer is 1234, write 234.



**Input:** Enter only the final three digits of the minimum sum.

The next term of a sequence is obtained by adding the same constant to the previous term. The 1st term is 12 and the 7th term is 66. What is the 50th term of this sequence?

**Input:** Choose one option.