AMO 2027 Grades 6-7 SET 21 - Self-study

The area of the shaded region is equal to $\frac17$ of the area of the big circle and is equal to $\frac15$ of the area of the small circle. The difference between the area of the big circle and the small circle is 18cm². What is the area of the small circle?


A. 36cm²

B. 72cm²

C. 90cm²

D. 126cm²

E. None of the above

**Editorial note for this self-study exercise:** Enter the numerical area in cm², without a unit or choice letter. The shaded region is the overlap shown in the original figure.

Let W be the smallest positive integer consisting of only digits 3 and 7 and divisible by both 3 and 7. What is the sum of digits of W?

**Editorial note for this self-study exercise:** Each digit may be 3 or 7; both different digits need not occur. Enter the digit sum, not W.

The expression below can be simplified to an integer. What is that integer?

$9700\times\left(\frac1{1\times4\times7}+\frac1{4\times7\times10}+\frac1{7\times10\times13}+\cdots+\frac1{94\times97\times100}\right)$

**Editorial note for this self-study exercise:** The first factors are 1, 4, 7, …, 94, increasing by 3. Enter the exact integer.

The ratio of Don’s money to Ahmad’s money is 4 : 7. The ratio of Don’s money to Sally’s money is 2 : 1. If they have $\$312$ altogether, how much money does Don have?

**Editorial note for this self-study exercise:** Enter Don’s amount in dollars, without a currency symbol. The source solution’s final label mistakenly calls Sally’s amount Don’s; the correction is explained below.

The figure below is made up of 4 rectangles P, Q, R, S. If the areas of rectangles P, Q, R are 48cm², 30cm² and 96cm² respectively, find the area of Rectangle S.

**Editorial note for this self-study exercise:** Use the aligned two-by-two rectangular partition in the original figure. Enter the area in cm² without a unit. The source key has a malformed unit; the issue is explained in the solution.

The original price of a ticket was $\$10$. When the price of the ticket was reduced, the number of customers increased by 50%, but the amount of money received only increased by 20%. What is the new reduced ticket price?


A. 1

B. 2

C. 3

D. 4

E. None of the above

**Editorial note for this self-study exercise:** Each customer buys one ticket at the stated price; no extra charges are included. Enter the new price in dollars without a currency symbol or choice letter.

A donkey and a horse have to carry a number of strong packages to the market.

Donkey: “Oh my God, the goods are too heavy! I’m going to be exhausted!”
Horse: “Why are you always complaining? You see, I carry two more packages than you! Besides, if you give me one of your packages, then the number of my packages will be twice of yours! Stop complaining!”

How many packages does the donkey carry?


A. 5

B. 7

C. 9

D. 11

E. None of the above

**Editorial note for this self-study exercise:** The question asks for the donkey’s number of packages before the proposed transfer. Enter that number.

A television set has a 30-inch screen size, i.e. the length of the diagonal of the screen is 30 inches. The aspect ratio of the screen is 4 : 3. Find the height of the screen. Leave your answer to the nearest whole number if necessary. (Note: The aspect ratio of the screen refers to the ratio of the width of the screen to its height.)


A. 13 inches

B. 18 inches

C. 21 inches

D. 24 inches

E. None of the above

**Editorial note for this self-study exercise:** Treat the screen as a rectangle. Enter the height in inches without a unit or choice letter.

Clara has $\frac45$ as much money as Sandra. Melanie has $\frac56$ as much money of Clara. If Sandra has $\$90$ more than Melanie, how much money does Clara have?

**Editorial note for this self-study exercise:** “5/6 as much money of Clara” is read as 5/6 of Clara’s money. Enter Clara’s amount in dollars without a currency symbol.

It is given that

$A=1.01\times1.19+1.02\times1.18+1.03\times1.17+\cdots+1.10\times1.10.$

Find the value of A. (Round off to the nearest whole number)

**Editorial note for this self-study exercise:** There are ten products: the first factor runs from 1.01 to 1.10 in steps of 0.01, while the second runs from 1.19 down to 1.10. Enter the rounded whole number.

Evaluate

$\left(\frac12+\frac13+\cdots+\frac1{20}\right)+\left(\frac23+\frac24+\cdots+\frac2{20}\right)$

$+\left(\frac34+\frac35+\cdots+\frac3{20}\right)+\cdots+\left(\frac{18}{19}+\frac{18}{20}\right)+\left(\frac{19}{20}\right).$

**Editorial note for this self-study exercise:** Every fraction i/j with whole numbers 1 ≤ i < j ≤ 20 occurs once. Enter an exact decimal or fraction; do not round.

In 2008, there were 21 states with 10 or more electoral votes, as shown in the table.

Electoral votes | Frequency
--- | ---
10 | 4
11 | 4
12 | 1
13 | 1
15 | 3
17 | 1
20 | 1
21 | 2
21 | 1
31 | 1
34 | 1
55 | 1

Based on the table, what was the median number of electoral votes for the 21 states?


A. 13

B. 15

C. 17

D. 20

E. None of the above

**Editorial note for this self-study exercise:** Use the table exactly as printed, including both rows labelled 21 (frequencies 2 and 1). This is a table-reading exercise, not a correction of historical data. Enter the median value, not a choice letter.

Five students took a quiz with a maximum score of 50. The scores of four of the students were 42, 43, 46, and 49. The score of the fifth student was N. The average of the five students’ scores was the same as the 3rd highest score. Find the number of values of N which are possible.


A. 3

B. 4

C. 1

D. 0

E. None of the above

**Editorial note for this self-study exercise:** A score lies between 0 and 50 inclusive. Rank all five scores, counting equal scores separately; the third highest is the middle score. Count distinct possible N values. The result does not require an integer-score assumption.

The highest common factor of two whole numbers is 20. The lowest common multiple of these two whole numbers is 200. If the sum of these two whole numbers is 140, what is the difference of these two whole numbers?


A. 60

B. 100

C. 120

D. 180

E. None of the above

**Editorial note for this self-study exercise:** The stated positive LCM excludes zero. Enter the nonnegative difference: larger number minus smaller number.

If the 7-digit number 123AB45 is divisible by 99, then what is A + B?


A. 3

B. 6

C. 9

D. 12

E. None of the above

**Editorial note for this self-study exercise:** A and B are single decimal digits, each from 0 to 9, and may be zero. Enter A + B, not the seven-digit number or a choice letter.