AMC8 2027 - Self-practice Set 6

How many integers from 1 through 46 are divisible by 3 or 5 orboth?

Define $a\otimes b=a\times b+a-b$. What is $11\otimes5$?

Two squares $ABCD$ and $PQRS$, each with side length $15$, overlap to form rectangle $AQRD$ with dimensions $15\times25$, as shown. What percentage of the area of rectangle $AQRD$ is shaded?

An eighth-grade class at Lincoln Middle School has $39$ students. Each owns a dog, a cat, or both. Twenty own dogs and twenty-six own cats. How many own both?

Jeff spins P, Q, R to choose the hundreds, tens, units digits of a three-digit number. How many numbers can he obtain?

Choose uniformly from all three-digit numbers whose digits belong to $\{1,3,5\}$. What is the probability of divisibility by 5?

Jane moves twice as fast as Hector around rectangle ABDE, where AB=3 and AE=6. They start at the midpoint of AE in the opposite directions shown. Where do they first meet?

Primes $p$, $q$ satisfyp=m+n andq=mn for positive integers $m$, $n$. Findp^q+q^p.

Define $a\diamond b=\frac{a\times b}{a+b}$. Find the value of $(3\diamond3)+(1\diamond2)$.

Rectangle $ABCD$ has $CD=3$ cm and $DA=5$ cm. Draw circles centered at $A$, $B$, and $C$ with radii $1$, $2$, and $3$ cm, respectively. Which value is closest to the area inside the rectangle but outside all three circles?

Kaleana shows her test score to Quay, Marty, and Shana, while each keeps their own score private. Quay thinks, "At least two of us have the same score." Marty thinks, "I do not have the lowest score." Shana thinks, "I do not have the highest score." Each thought is a conclusion known with certainty based only on that student's own score and Kaleana's score. In what order do Marty (M), Quay (Q), and Shana (S) rank from lowest to highest?

Two fair dice are rolled. How many ordered outcomes have a sum divisible by 3?

Fair spinners show 1, 3, 5 and 2, 4, 6. What is the probability that their sum is prime?

Two 600 ml containers are 1/3 and 2/5 full of orange juice. Each is topped up with water, then both are poured into a larger container. What fraction is orange juice?

Find the units digit of 1997^1997−1.

Evaluate $99-97+95-93+\ldots+3-1$.

Let $A$ be the area of a triangle with side lengths $25,25,30$, and let $B$ be the area of a triangle with side lengths $25,25,40$. Which of the following correctly relates $A$ and $B$?

Alice has $9$ identical-looking coins. Eight have the same weight, and one counterfeit coin is heavier. What is the minimum number of weighings on a two-pan balance needed to identify the counterfeit coin?

Distribute 6 identical pencils among 3 distinct friends, each receiving at least 1. How many distributions are possible?

The graph gives Alice study times 60, 90, 100, 80, 70 minutes, and Jacky 70, 80, 120, 110, 50 minutes, on five weekdays. How many more minutes per day does Jacky study on average?