AMC8 2027 - Self-practice Set 4

Last year Ara and Shea had the same height. Shea grows 20%; Ara grows by half as much as Shea. Shea is now 60 inches tall. How tall is Ara now?

Alice,Betsy,Casey run a lap in 141/10,235/14,94/7 seconds respectively. They start together. How many laps has Alice run when all three first return together to the start?

Evaluate the following expression.
$$1+3+5+\ldots+2019-2-4-6-\ldots-2018.$$

In triangle $ABC$, point $D$ lies on side $AC$ such that $BD=DC$ and $\angle BCD=70^\circ$. Find $\angle ADB$.

Six athletes play a tournament in which each pair meets exactly once. Helen wins $4$ games, Ines $3$, Janet $2$, Kendra $2$, and Lara $2$. How many games does Monica win if there are no draws?

Color a $2\times2$ grid gray or white. No gray cell may be below or to the left of a white cell in the same column or row. How many colorings are possible, including all-gray and all-white?

Four friends randomly occupy the four sides of a square table. What is the probability that Alice and Charlie sit opposite each other?

Alex, Besty, and Cathy share apples. Alex takes one quarter of all apples plus 3. Besty takes one third of the remainder plus 2. Cathy receives the remaining 8. How many apples were there?

What is the units digit of 13^2003?

Evaluate the following expression.
$$1+\frac1{1+2}+\frac1{1+2+3}+\ldots+\frac1{1+2+\ldots+2019}$$

Divide the circle centered at $O$ into $12$ equal arcs, labeled $A$ through $L$, as shown. How many degrees is $x+y$?

Choose any three nonnegative integers. Which statement must be true?

A palindrome reads the same forwards and backwards. A 12-hour digital clock displays hours without a leading zero and two-digit minutes. Ignoring the colon, how many palindromic times occur in a 12-hour period?

A spinner has regions A, B, C, with probabilities 1/3 and 1/2 for A and B. What is the probability for C?

A rectangle becomes 20% longer and 50% wider. By what percent does its area increase?

Find the digit sum of the product of a 94-digit all-nine number and a 94-digit all-four number.

Define $a*b=ab-a-b+2$. Given that $7*y=13$, find $y$.

A circle of diameter $1$ is inscribed in a square and circumscribed about another square, as shown. Which fraction is closest to the ratio of the shaded area inside the circle to the area between the two squares?

Lý Thông, Thạch Sanh, and a Knight bring the head of Xà Tinh to the king. The king asks, "Who killed Xà Tinh?"
Lý Thông says, "I killed Xà Tinh."
Thạch Sanh says, "The Knight killed Xà Tinh."
The Knight says, "Lý Thông did not kill Xà Tinh."
Exactly one of the three is lying. Who killed Xà Tinh?

How many three-element subsets of $\{1,2,3,4,5,6,7,8,9\}$ contain 5 and have sum 15?