AMC8 2027 - Self-practice Set 3

In the diagram, triangles $ABC$, $ADE$, and $EFG$ are equilateral. Points $D$ and $G$ are the midpoints of $AC$ and $AE$, respectively. Find the perimeter of polygon $ABCDEFG$ if $AB=4$.

Abby, Bret, Carl, and Dana sit in a row of seats numbered $1$ through $4$. Joe says, "Bret sits next to Carl" and "Abby sits between Bret and Carl." Both statements are false. Bret sits in seat $3$. Who sits in seat $2$?

How many inequivalent ways are there to color two cells of a $3\times3$ grid, identifying patterns under rotation or reflection?

In the illustrated survey of 650 students, 250 prefer seafood pizza and 100 sausage pizza. What is the seafood-to-sausage ratio?

A\$100 shirt is discounted 20%, then\$5 with a coupon. Sales tax is 8% of the final price. What does the customer pay?

What is the remainder of 1492·1776·1812·1996 modulo 5?

When computing $12+5\times3$, Alice forgets that multiplication and division precede addition and subtraction. She adds first and obtains $A$. Let $B$ be the correct value. What is $A-B$?

Inside square $ABCD$ are three smaller squares with the labeled side lengths. What is the area of the shaded L-shaped region?

Five cards lie on a table as shown. Each has a letter on one side and a nonnegative integer on the other. Jane claims, "If a card has a vowel on one side, its other side has an even number." Mary can disprove Jane by turning over one card. Which card should Mary turn over?

How many isosceles triangles have positive integer side lengths and perimeter 23?

A KAMC Radio survey has 200 respondents, 136 listeners, and 58 female listeners. What percentage of all respondents are male listeners?

Antonette answers 70% of 10 questions,80% of 20 questions, and 90% of 30 questions correctly. What percentage of the combined 60-question test does she answer correctly?

How many digits does 4^5·5^10 have?

If $a\otimes b=\frac{a+b}{a-b}$, what is $(6\otimes5)\otimes4$?

If $\angle A=20^\circ$ and $\angle AFG=\angle AGF$, what is $\angle B+\angle D$?

Hoàng and Đức have three hats: two white and one black. While blindfolded, each wears one hat, and someone secretly puts away the unused hat. They take turns opening their eyes, seeing the other person's hat, and trying to identify their own hat.
Hoàng opens his eyes first but cannot determine his own hat color. Đức then says, "I know the color of my hat without opening my eyes."
Both are excellent reasoners. What can be concluded?

What is the minimum number of occupied seats in a row of 120 that forces every new arrival to sit next to someone?

Abe has 1 blue and 1 red marble; Bae has 1 blue, 1 yellow, and 2 red marbles. Each independently picks a random marble. What is the probability that their colors match?

A coat and shirt originally cost\$80 and\$40, respectively. They are discounted 40% and 55%. What percentage of the total original price does Chris save?

Students sit around a round table and pass a bag of 100 candies, each taking one before passing it on. Chris takes both the first and last candy. Which could be the number of students?