AMO 2027 Grades 6-7 SET 25 - Self-study

The counting numbers are arranged five at a time in every row, as shown. What is the least counting number in the row in which 214 will eventually appear?



**Input:** Choose one option.

The numbers 1 through 12 are placed in the diagram, one in each circle, so that the sum of the numbers along each side of the triangle is the same. What is the largest possible value of this sum per side?



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

A triangle has sides 35 cm, 37 cm and 12 cm. Identify the type of triangle.

**Input:** Choose one option.

Find the ones digit of $2^{10}+3^9$.

**Input:** Choose one option.

How many three-digit natural numbers $n$ satisfy all three conditions: $n-14$ is divisible by 7; $n-24$ is divisible by 8; and $n-36$ is divisible by 9?

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

What is the sum of the prime numbers strictly between 2000 and 2021?

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

Andy and Brett begin at opposite points on a circular track of circumference 300 feet and jog in the directions shown. Andy jogs at 4.3 feet per second and Brett at 5.5 feet per second. How many seconds will it take for them to meet for the first time?



**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.

Find the largest four-digit number that leaves a remainder of 3 when divided by each of 4, 7 and 13.

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

How many shortest paths along a 3 × 3 grid lead from one corner to the diagonally opposite corner?

Editorial clarification: use 3 rows and 3 columns of vertices, joined horizontally and vertically, as illustrated. This is a grid of 2 × 2 square cells. Travel from A to B along the drawn segments.



**Input:** Choose one option.

$2^{400}$ can be written as a repeated product of the number 16: $\underbrace{16\times16\times16\times\cdots\times16}_{n\text{ factors}}$. What is the value of $n$?

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

Nicholas writes the positive integers from 1 to 50 in an ordered list. On Monday he crosses out every multiple of 2; on Tuesday every multiple of 3; on Wednesday every multiple of 5; on Thursday every multiple of 7; on Friday every multiple of 11; and on Saturday every multiple of 13. How many of the original 50 integers are not crossed out?

**Input:** Choose one option.

A palindrome reads the same forwards and backwards. The number 2017102 is a 7-digit palindrome. Let $A$ be the least palindrome greater than 2017102, and $B$ the greatest palindrome less than 2017102. Find $(A-B)/10$.

**Input:** Choose one option.

For how many whole numbers from 5001 to 5499 does the product of the middle two digits exceed 6?

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

Two prime numbers have product 9991. Find the smaller prime.

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