2022 AMC 8 Problems
IMPORTANT: THESE ARE NOT THE 2022 AMC 8 PROBLEMS. THIS IS COPY PASTED FROM THE 2020 AMC 8 PROBLEMS WIKI PAGE.
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
Problem 1
The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches?
usepackage("mathptmx"); defaultpen(linewidth(0.5)); size(5cm); defaultpen(fontsize(14pt)); label("$\textbf{Math}$", (2.1,3.7)--(3.9,3.7)); label("$\textbf{Team}$", (2.1,3)--(3.9,3)); filldraw((1,2)--(2,1)--(3,2)--(4,1)--(5,2)--(4,3)--(5,4)--(4,5)--(3,4)--(2,5)--(1,4)--(2,3)--(1,2)--cycle, mediumgray*0.5 + lightgray*0.5); draw((0,0)--(6,0), gray); draw((0,1)--(6,1), gray); draw((0,2)--(6,2), gray); draw((0,3)--(6,3), gray); draw((0,4)--(6,4), gray); draw((0,5)--(6,5), gray); draw((0,6)--(6,6), gray); draw((0,0)--(0,6), gray); draw((1,0)--(1,6), gray); draw((2,0)--(2,6), gray); draw((3,0)--(3,6), gray); draw((4,0)--(4,6), gray); draw((5,0)--(5,6), gray); draw((6,0)--(6,6), gray); (Error making remote request. Unexpected URL sent back)
Problem 2
Consider these two operations:
What is the value of
Problem 3
When three positive integers ,
, and
are multiplied together, their product is
. Suppose
. In how many ways can the numbers be chosen?
Problem 4
Problem 5
Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned years old, she received a newborn kitten as a birthday present. Today the sum of the ages of the two children and the kitten is
years. How many years older than Bella is Anna?
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
In how many ways can the letters in BEEKEEPER be rearranged so that two or more Es do not appear together?
Problem 15
Problem 16
Problem 17
How many factors of have more than
factors? (As an example,
has
factors, namely
and
)
Problem 18
Rectangle is inscribed in a semicircle with diameter
as shown in the figure. Let
and let
What is the area of
Problem 19
A number is called flippy if its digits alternate between two distinct digits. For example, and
are flippy, but
and
are not. How many five-digit flippy numbers are divisible by
Problem 20
A scientist walking through a forest recorded as integers the heights of trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
Problem 21
A game board consists of squares that alternate in color between black and white. The figure below shows square
in the bottom row and square
in the top row. A marker is placed at
A step consists of moving the marker onto one of the adjoining white squares in the row above. How many
-step paths are there from
to
(The figure shows a sample path.)
Problem 22
When a positive integer is fed into a machine, the output is a number calculated according to the rule shown below.
For example, starting with an input of the machine will output
Then if the output is repeatedly inserted into the machine five more times, the final output is
When the same
-step process is applied to a different starting value of
the final output is
What is the sum of all such integers
Problem 23
A or
is placed in each of the nine squares in a 3-by-3 grid. Shown below is a sample configuration with three
s in a line.
[center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC8wL2ZjMDdkOGRlMzVjNWZlOWE3NTI5MjkwODIxODQ0YWRkYmQxZmJiLnBuZw==&rn=dGljX3RhY190b2VfdHJpYW5nbGVzLnBuZw==[/img][/center]
How many configurations will have three s in a line and three
s in a line?
Problem 24
A large square region is paved with gray square tiles, each measuring
inches on a side. A border
inches wide surrounds each tile. The figure below shows the case for
. When
, the
gray tiles cover
of the area of the large square region. What is the ratio
for this larger value of
Problem 25
Rectangles and
and squares
and
shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of
in units?