AMO 2027 Grade 7 SET 1 - Self-study

Thirty-six points form a unit-square array as shown. Figure ABCDE consists entirely of straight-line segments with vertices A, B, C, D and E. Find the sum of its interior angles, in degrees.



**Input:** Choose one option.

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.

Some of the 12 edges of a cube are coloured red so that each face has exactly 2 red edges. How many edges are red?

**Input:** Choose one option.

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

**Input:** Choose one option.

$\overline{XYX}$ and $\overline{YXY}$ are three-digit integers with distinct nonzero digits $X,Y$. How many different values can the sum $\overline{XYX}+\overline{YXY}$ have?

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

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

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

Given $\overline{XYZ}+\overline{ZYX}=\overline{ABAA}$, where $A,B,X,Y,Z$ are five different positive single-digit numbers, how many ordered pairs $(X,Z)$ satisfy the equation? Swapping $X$ and $Z$ counts as a different pair.



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

Mel sells jumbo cheesy pretzel nuggets in a 4-nugget “diet” size, a 7-nugget “share” size and a 17-nugget “party” size. What is the greatest total number of nuggets that cannot be purchased? For example, 8 nuggets can be purchased, but 9 cannot. Assume any nonnegative whole number of each pack may be bought, and packs cannot be split.

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

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.

Let $N$ be the greatest possible negative quotient when two distinct integers chosen from $\{-7,-5,-4,-3,-2,0,1,2,6\}$ are divided. Find $213+245N$.

**Input:** Choose one option.

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.

A survey of 40 students at Mathlete Academy found that 27 students play sports and 20 students play an instrument. If 8 students in the survey do not play a sport and do not play an instrument, how many students play both a sport and an instrument?

**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.

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

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

Find the greatest positive integer that divides 266, 516 and 741 with the same remainder in each case.

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