AMC8 2027 - Self-practice Set 7

How many axis-aligned $2\times2$ squares can be cut from a $5\times5$ grid?

A 60-second traffic-light cycle has 25 seconds green, 5 yellow, and 30 red. At a uniformly random arrival time, what is the probability of having to stop at yellow or red?

A car travels 80 miles in 1.5 hours, rests 30 minutes, and travels 100 miles in 2 hours. What is its average speed for the entire trip?

What is the units digit of the product of 6 consecutive nonnegative integers?

Evaluate the following expression.
$$H=\frac1{2\times4}+\frac1{4\times6}+\frac1{6\times8}+\ldots+\frac1{2018\times2020}.$$

Three arcs of radius $5$ enclose the region shown. Arcs $AB$ and $AD$ are quarter circles, and arc $BCD$ is a semicircle. Find the area enclosed by the three arcs.

Five friends compete at darts. Each throws two darts at a board with ten scoring rings worth $1$ through $10$ points. All ten darts land in different scoring rings. Alice scores $16$, Ben $4$, Cindy $7$, Dave $11$, and Ellen $17$. Who hits the $6$-point ring?

A $3\times3\times3$ cube is colored as shown, with opposite faces matching. What is the minimum number of unit cubes with at least one gray face?

Tyler caught 2, 0, 1, 3, 0, 3, 3, 1, 2 fish on nine evenings. Which ordering of mean, median, and mode is correct?

Tori has a 75-question test:10 arithmetic,30 algebra,35 geometry. She gets 70%,40%,60% correct in those sections. How many additional correct answers are needed to reach 60% overall?

How many positive divisors of 432 are perfect squares?

What is the value of the following expression?
$$\left(1+\frac11\right)\left(1+\frac12\right)\left(1+\frac13\right)\left(1+\frac14\right)\left(1+\frac15\right)\left(1+\frac16\right)$$

Triangle $ABC$ is right-angled at $B$. Each side is the diameter of a semicircle, as shown. The semicircle with diameter $AB$ has area $8\pi$, and the semicircular arc with diameter $AC$ has length $8.5\pi$. What is the radius of the semicircle with diameter $BC$?

Alan, Beth, Carlos, and Diana discuss their mathematics test grades.
Alan says, "If I received an A, then Beth also received an A."
Beth says, "If I received an A, then Carlos also received an A."
Carlos says, "If I received an A, then Diana also received an A."
All statements are true, but exactly two students received A grades. Which two?

Bicycle plates choose one letter from sets of sizes 5, 3, 4. Two new letters can be added to one set or split between two sets. What is the maximum increase in possible plates?

A class survey has 6, 8, 4, 2, 5 students tasting candy types A, B, C, D, E, respectively. What percentage chose E?

Last week a store sold 4 cans of sardines for\$5. This week it sells 5 cans for\$4. By approximately what percent has the price per can decreased?

Three primes sum to 40. What is the greatest possible product of two of them?

Evaluate the following expression.
$$\frac{2019^2-\left(2017^2-\left(2015^2-\left(2013^2-\left(\ldots-\left(3^2-1^2\right)\right)\right)\right)\right)}{1010}$$

Isosceles trapezoid $ABCD$ has $AB=CD$ and the dimensions shown. What is its perimeter?