2022 AMC 8

Solution 1 (A quick solution)

We can see that there are 4 whole squares, since the area of each square will be , . Next, there are half squares, and half squares is whole square, so whole squares. The area of this will be . Finally, we can add the numbers: .

Solution 2

Draw the following four lines as shown:

We see these lines split the figure into five squares with side length . Thus, the area is

We can find a general solution to any .

Solution 1

The positive divisors of are

It is clear that so we apply casework to

- If then

- If then

- If then

- If then

Together, the numbers and can be chosen in ways.

Solution 2

The positive divisors of are

We apply casework to :

If , then there are cases:

-

-

-

If , then there is only case:

-

In total, there are ways to choose distinct positive integer values of .

When M is first reflected over the line we obtain the following diagram:

When M is then reflected over the line we obtain the following diagram:

Therefore, the answer is

Five years ago, Bella was years old, and the kitten was years old.

Today, Bella is years old, and the kitten is years old. It follows that Anna is years old.

Therefore, Anna is years older than Bella.

Solution 1

Let the smallest number be It follows that the largest number is

Since and are equally spaced on a number line, we have

Solution 2

Let the common difference of the arithmetic sequence be . Consequently, the smallest number is and the largest number is . As the largest number is times the smallest number, . Finally, we find that the smallest number is .

Solution 1

Notice that the number of kilobits in this song is

We must divide this by in order to find out how many seconds this song would take to download:

Finally, we divide this number by because this is the number of seconds to get the answer

Solution 2

We seek a value of that makes the following equation true, since every other quantity equals .

Solving yields .

Solution 1

Note that common factors (from to inclusive) of the numerator and the denominator cancel. Therefore, the original expression becomes

Solution 2

The original expression becomes

Initially, the difference between the water temperature and the room temperature is degrees Fahrenheit.

After minutes, the difference between the temperatures is degrees Fahrenheit.

After minutes, the difference between the temperatures is degrees Fahrenheit.

After minutes, the difference between the temperatures is degrees Fahrenheit. At this point, the water temperature is degrees Fahrenheit.

Remark:

Alternatively, we can condense the solution above into the following equation:

Solution 1 (Analysis) Note that:

- At Ling's car was miles from her house.

- From to Ling drove to the hiking trail at a constant speed of miles per hour.

It follows that at Ling's car was miles from her house.

- From to Ling did not move her car.

It follows that at Ling's car was still miles from her house.

- From Ling drove home at a constant speed of miles per hour. So, she arrived home hour later.

It follows that at Ling's car was miles from her house.

Therefore, the answer is

Solution 2 (Elimination)

Ling's trip took hours, thus she traveled for miles. Choices , , and are eliminated. Ling drove miles per hour (mph) to the mountains, and mph back to her house, so the rightmost slope must be steeper than the leftmost one. Choice is eliminated. This leaves us with .

Solution 1

Let's say that the first strand of pasta he had is . The first time he takes a bite of this strand will make it pieces. This is . The second time he takes the bites of EACH strand will become . The is for the two bites he took, for each strand. This equation is showing that for every bite that he is taking, he is taking off inches. This is being showed in the . He took bites, each taking off inches. Now, the problem is stating that their are pieces of pasta and the TOTAL length is inches. We have to plug in the total number of bites to the equation . To find this out, we can see that for the number of strands there are, he took less bite than that number. For example, if there are pieces, he took bite to get those. Since there are pieces, he took bites. The equation then becomes, . To solve this, . Therefore, the answer is .

Solution 2

If there are pieces of pasta, Henry took bites. Each of these bites took inches of pasta out, so all his bites in total took away inches of pasta. Thus, the original piece of pasta was inches long.

Solution 1

First, we calculate that there are a total of possibilities. Now, we list all of two-digit perfect squares. and are the only ones that can be made using the spinner. Consequently, there is a probability that the number formed by the two spinners is a perfect square.

Solution 2

There are total possibilities of . We know , which is a number from spinner , and is a number from spinner . Also, notice that there are no perfect squares in the s or s, so only values of N work, namely and . Hence, .

Let and be positive integers such that and It follows that for some positive integer We wish to find the number of possible values for

By substitution, we have from which Note that each generate a positive integer for so there are possible values for

All valid arrangements of the letters must be of the form

The problem is equivalent to counting the arrangements of and into the four blanks, in which there are ways.

Solution 1 

We are looking for a black point, that when connected to the origin, yields the lowest slope. The slope represents the price per ounce. We can visually find that the point with the lowest slope is the blue point. Furthermore, it is the only one with a price per ounce significantly less than . Finally, we see that the blue point is in the category with a weight of ounces.

Solution 2 (Elimination)

By the answer choices, we can disregard the points that do not have integer weights. As a result, we obtain the following diagram:

We then proceed in the same way that we had done in Solution . Following the steps, we figure out the blue dot that yields the lowest slope, along with passing the origin. We then can look at the x-axis(in this situation, the weight) and figure out it has ounces.

Solution 1 (Arithmetic)

Note that the sum of the first two numbers is the sum of the middle two numbers is and the sum of the last two numbers is

It follows that the sum of the four numbers is so the sum of the first and last numbers is Therefore, the average of the first and last numbers is

Solution 2 (Algebra)

Let and be the four numbers in that order. We are given that

and we wish to find

We add and then subtract from the result:

Solution 1

Notice that once the units digit of will be because there will be a factor of Thus, we only need to calculate the units digit of

We only care about units digits, so we have which has the same units digit as The answer is

Solution 2 (Solution 1 worded differently)

We can see that after in the sequence, the units digit is always (every value after is a multiple of ). Therefore, our answer is the sum of the units digits of and respectively. This sum is equal to , or

Solution 1

The midpoints of the four sides of every rectangle are the vertices of a rhombus whose area is half the area of the rectangle: Note that the diagonals of the rhombus have the same lengths as the sides of the rectangle.

Let and Note that and are the vertices of a rhombus whose diagonals have lengths and It follows that the dimensions of the rectangle are and so the area of the rectangle is

Solution 2

If a rectangle has area then the area of the quadrilateral formed by its midpoints is

Define points and as Solution 1 does. Since and are the midpoints of the rectangle, the rectangle's area is Now, note that is a parallelogram since and As the parallelogram's height from to is and its area is Therefore, the area of the rectangle is

We notice that students have scores under currently and only have scores over . We find the median of these two numbers, getting:

Thus, we realize that students must have their score increased by .

So, the correct answer is .

Solution 1

The sum of the numbers in each row is . Consider the second row. In order for the sum of the numbers in this row to equal , the two shaded numbers must add up to :

If two numbers add up to , one of them must be at least : If both shaded numbers are no more than , their sum can be at most . Therefore, for to be larger than the three missing numbers, must be at least . We can construct a working scenario where :

So, our answer is .

Solution 2

The sum of the numbers in each row is and the sum of the numbers in each column is

Let be the number in the lower middle. It follows that or

We express the other two missing numbers in terms of and as shown below:

We have and Note that the first inequality is true for all values of We only need to solve the second inequality so that the third inequality is true for all values of By substitution, we get from which

Therefore, the smallest possible value of is

Solution 1 (Inequalities)

Let be the number of shots that Candace made in the first half, and let be the number of shots Candace made in the second half. Since Candace and Steph took the same number of attempts, with an equal percentage of baskets scored, we have In addition, we have the following inequalities:

and

Pairing this up with we see the only possible solution is for an answer of

Solution 2 (Answer Choices)

Clearly, Steph made shots in total. Also, due to parity reasons, the difference between the amount of shots Candace made in the first and second half must be odd. Thus, we can just test 7, 9, and 11, and after doing so we find that the answer is

Solution 1

Initially, suppose that the bus is at Stop (starting point) and Zia is at Stop

We construct the following table of -minute intervals:

Note that Zia will wait for the bus after minutes, and the bus will arrive minutes later.

Therefore, the answer is

Solution 2

Since Zia will wait for the bus if the bus is at the previous stop, we can create an equation to solve for when the bus is at the previous stop. The bus travels of a stop per minute, and Zia travels of a stop per minute. Now we create the equation, (the accounts for us wanting to find when the bus reaches the stop before Zia's). Solving, we find that Now Zia has to wait minutes for the bus to reach her, so our answer is

Solution 1 (Casework)

Notice that diagonals and a vertical-horizontal pair can never work, so the only possibilities are if all lines are vertical or if all lines are horizontal. These are essentially the same, so we'll count up how many work with all lines of shapes vertical, and then multiply by at the end.

We take casework:

Case : lines: In this case, the lines would need to be of one shape and of another, so there are ways to arrange the lines and ways to pick which shape has only one line. In total, this is

Case : lines: In this case, the lines would be one line of triangles, one line of circles, and the last one can be anything that includes both shapes. There are ways to arrange the lines and ways to choose the last line. (We subtract from the last line because one arrangement of the last line is all triangles and the other arrangement of the last line is all circles, which causes Case to overlap with Case and further complicating the solution.) In total, this is