AMO 2027 Grades 6-7 SET 29 - Self-study

The length of each edge of a cube is increased by 10%. By what percent is its volume increased?

**Input:** Choose one option.

How many three-digit natural numbers are divisible by 5? Natural numbers here are positive integers starting from 1.

**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.

Lean cycles at constant speed along a straight street from A to B. After starting at A, he is $\frac5{17}$ of the way to B when he hears a car approaching from behind at 391 km/h. If he immediately turns back toward A, he meets the car at A; if he continues toward B, the car overtakes him at B. How fast does Lean cycle, in km/h? Assume both speeds stay constant and turning takes no time.

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

Using the digits 3, 4, 5, 6, 7 exactly once each, form the largest and the smallest numbers. What is the difference between the place values of the digit 7 in these two numbers?

**Input:** Choose one option.

A number has 2021 digits. Let P be the sum of its digits, Q the sum of the digits of P, and R the sum of the digits of Q. Each operation is one ordinary sum of digits, not a repeated digital-root calculation. Find the largest possible Q.

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

The prime factorization of 2016 is $2^5\times3^2\times7$, and that of 2018 is $2\times1009$. Find the number of distinct positive divisors of $2016\times2018$.

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

Find all ordered pairs of integers $(x,y)$ satisfying $x^y=16$. Sum the x-coordinates of all those pairs and separately sum their y-coordinates, counting one coordinate per pair. Find the positive difference between these two sums.

**Input:** Choose one option.

Lucy is Sherry’s mother. Their current ages are two-digit whole numbers with the same digits in reverse order. In 13 years, Lucy will be twice as old as Sherry. How old will Lucy be when Sherry reaches Lucy’s current age?

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

The surface areas of two spheres are in the ratio 4:9. Find the ratio of their volumes, in the same order.

**Input:** Choose one option.

Use all nine digits 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once across the numerator and denominator to create a fraction equal to $\frac13$. What is the numerator? The conditions do not determine a unique numerator; enter any one valid numerator.

**Input:** Enter one valid numerator as an integer, without the denominator.

What is the value of $(115+114+113+112+111)-(101+102+103+104+105)$?

**Input:** Choose one option.

Simplify and round to the nearest whole number: $$\frac{16}{\sqrt{14}+\sqrt{10}}-\frac{58}{3\sqrt5-4}+\sqrt{180}+\sqrt{160}.$$

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

Stacy has 53 marbles and puts them into three nonempty piles, each containing an odd number of marbles. How many different ways can she do this? The order of the piles does not matter: 1, 1, 51 is the same arrangement as 1, 51, 1.

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

A two-digit number is 6 more than 7 times the sum of its digits. Its tens digit is 2 more than twice its ones digit. How many such two-digit numbers are there?

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