AMC8 2027 - Self-practice Set 8

Evaluate $\frac{3+6+9+\ldots+75}{4+8+12+\ldots+100}$.

Rectangle $ABCD$ has area $72$. Points $M$ and $N$ are the midpoints of $BC$ and $CD$, respectively. Find the area of triangle $AMN$.

A jar contains red, blue, and yellow marbles. All but six are red, all but eight are blue, and all but four are yellow. How many marbles are in the jar?

Pick one candy numbered 1, 3, 5 from bag A and one numbered 2, 4, 6 from bag B. How many distinct sums are possible?

Jacky collects stamps from Brazil, France, Peru, Spain. Each Brazilian or French stamp costs 6 cents, each Peruvian stamp 4 cents, and each Spanish stamp 5 cents. The numbers for decades 50 s, 60 s, 70 s, 80 s are Brazil(4, 7, 12, 8), France(8, 4, 12, 15), Peru(6, 4, 6, 10), Spain(3, 9, 13, 9). Brazil and Peru are South American; France and Spain are European. Which choice is closest to the average cost per 70 s stamp?

Today is Sunday. What weekday is 5000 days from today?

Positive integers $m$, $n$ satisfy 11|(5 m+3 n). What is the remainder of 9 m+n modulo 11?

Use every digit $0$ through $9$ exactly once to form two five-digit numbers with the largest possible sum. Which number could be one of them?

An artist stacks 14 unit cubes on the floor as shown, then paints all exposed surfaces except the floor-contact area. How many square metres does she paint?

Jacky collects stamps from Brazil, France, Peru, Spain. Each Brazilian or French stamp costs 6 cents, each Peruvian stamp 4 cents, and each Spanish stamp 5 cents. The numbers for decades 50 s, 60 s, 70 s, 80 s are Brazil(4, 7, 12, 8), France(8, 4, 12, 15), Peru(6, 4, 6, 10), Spain(3, 9, 13, 9). Brazil and Peru are South American; France and Spain are European. How many European stamps from the 80 s did he collect?

A car leaves at 40 miles/hour. Two hours later another follows at 60 miles/hour. How long does the second car take to catch the first?

Two 99-digit numbers alternate 3,0 and 5,0 respectively, starting and ending in 3 and 5. If their product has thousands digit A and units digit B, find A+B.

A special calculator has only two buttons, $[+1]$ and $[\times2]$. Pressing a button immediately updates the displayed number. For example, starting at $9$, pressing $[+1]$ displays $10$, then pressing $[\times2]$ displays $20$. What is the minimum number of presses needed to reach $200$ starting from $1$?

The figure consists of 12 unit squares. In how many ways can a T-shaped tetromino be cut from it?

Karl buys 5 notebooks at\$2.50 each. The next day all items are 20% off. How much would he save by buying them the next day?

Nonnegative integers $x$, $y$, $z$ satisfy 11|(7 x+2 y−5 z). Find the remainder of 3 x−7 y+12 z modulo 11.

The numbers $1$ through $2021$ are written on a board. Ali erases any two numbers and writes their difference. He repeats until exactly one number remains.
a) How many operations does he perform?
b) Can the final number be $2$?

An auditorium has 20 rows with 10, 11,…, 29 seats. No two students may sit next to one another in a row. What is the maximum number of students?

Ana earns\$2000 in May. Her salary increases 20% in June and decreases 20% in July. What is her monthly salary after both changes?

In how many ways can 47 be expressed as the sum of two primes?