2020 AMC 8 Exam Problems
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All of the real AMC 8 and AMC 10 problems in our complete solution collection are used with official permission of the Mathematical Association of America (MAA).
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1.
Luka is making lemonade to sell at a school fundraiser. His recipe requires times as much water as sugar and twice as much sugar as lemon juice. He uses cups of lemon juice. How many cups of water does he need?
Answer: E
Video solution:
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Written solution:
Since Luka needs twice as much sugar as lemon, he needs cups of sugar. Since Luka also needs times as much water as sugar, he needs cups of water.
Thus, the correct answer is E.
2.
Four friends do yardwork for their neighbors over the weekend, earning $15, $20, $25, and $40 respectively. They decide to split their earnings equally among themselves. In total how many dollars will the friend who earned $40 give to the others?
Answer: C
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Written solution:
First, the total amount of money that they make is $15+$20+$25+$40=$100.
Since they divide this equally, they each get dollars.
This means that the person who earned $40 earned $15 more than what he will end up with, so he gives $15 to the others.
Thus, the correct answer is C.
3.
Carrie has a rectangular garden that measures feet by feet. She plants the entire garden with strawberry plants. Carrie is able to plant strawberry plants per square foot, and she harvests an average of strawberries per plant. How many strawberries can she expect to harvest?
Answer: D
Video solution:
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Written solution:
First, the size of the garden is
Next, since there are plants per square foot, Carrie can plant plants total.
Finally, since there are strawberries per plant, Carrie can harvest strawberries total.
Thus, the correct answer is D.
4.
Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?
Answer: B
Video solution:
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Written solution:
Firstly, let's try and find a pattern for the number of dots in the th ring. Notice that for the th ring, there is always dots on each edge of the hexagon. However, this overcounts as each vertex of the hexagon will be counted twice.
Therefore, we can claim that for the th ring, the number of dots is equal to Note that this pattern does not hold for
Therefore, for the first hexagon, we have dot, the 2nd hexagon adds dots, the 3rd hexagon adds dots, as reflected in the diagram. Extending the pattern, we can say that the fourth hexagon adds dots.
Finally, notice that the total number of dots in the hexagon is equal to the sum of all the rings up to the th ring. This is the same as saying: Therefore, for we have a total of
Thus, the correct answer is B.
5.
Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of cups. What percent of the total capacity of the pitcher did each cup receive?
Answer: C
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Written solution:
Since we start with of the pitcher, we have of the pitcher full as
Now, since cups each have the same amount of juice, they each have one-fifth of the
This means they have
Thus, the correct answer is C.
6.
Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?
Aaron
Darren
Karen
Maren
Sharon
Answer: A
Video solution:
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Written solution:
We know that they must be arranged as such: \text{__, __, __, __, Maren}, with each "__" representing people whose location we don't yet know.
Note that Maren must be there since he sat in the last car.
Since we know Sharon sat in front of Aaron, we can take cases of the possible locations of them both.
Case 1: \text{Sharon, Aaron, __, __, Maren} Since Darren sits in front of Aaron, we can't find a space for him so this case can't work.
Case 2: \text{__, Sharon, Aaron, __, Maren} Since Darren sits in front of Aaron, we know the configuration must be Darren, Sharon, Aaron, __, Maren -- leaving one spot for Karen.
The final configuration is Darren, Sharon, Aaron, Karen, Maren.
Case 3: \text{__, __, Sharon, Aaron, Maren} This case would involve Darren and Karen being next to each other, which contradicts the condition that at least one person is between them.
Since the only valid configuration is Darren, Sharon, Aaron, Karen, Maren, we know Aaron is in the middle.
Thus, the correct answer is A.
7.
How many integers between and have four distinct digits arranged in increasing order? (For example, is one integer.)
Answer: C
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Written solution:
Since the integers are between and we know the thousands digit must be 2 and the hundreds digit must be between and
Since the digits are increasing, the second digit must be greater than so it can only be
This means the tens and units digits are different digits each greater than or equal to
Suppose we choose 2 distinct digits each greater than or equal to There are digits for the first choice and digits for the second choice.
This means there are combinations.
However, we ignore exactly half of these combinations as each combination has an equal likelihood of being ascending or descending.
This leaves combinations.
Thus, the correct answer is C.
8.
Ricardo has coins, some of which are pennies (-cent coins) and the rest of which are nickels (-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least possible amounts of money that Ricardo can have?
Answer: C
Video solution:
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Written solution:
Let be the number of pennies Ricardo has and let be the number of nickels he has.
We know that and by the problem statement.
This means Therefore, It follows, then, that Ricardo has cents.
Therefore, to maximize the money he has, we maximize the number of nickels he has, and minimizing the money he has involves minimizing the number of nickels he has. The maximum number of nickels he can have is so he can have at most cents. The minimum number of nickels he can have is so he has at most cents. The difference between the maximum and minimum amount of money he can have is:
Thus, the correct answer is C.
9.
Akash's birthday cake is in the form of a inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into smaller cubes, each measuring inch, as shown below. How many small pieces will have icing on exactly two sides?
Answer: D
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Written solution:
On the top side, there are cubes that are only on the top. Also, there are cubes that have icing on sides. Therefore, we have cubes on the top that have icing on exactly sides.
For the other sides, we need to find the number of cubes that have icing on exactly two sides, exluding the cubes we counted on the sides. Each face has cubes on the edge that have icing on exactly two sides. However, this would double count when taking into account the other faces, as each cube would be counted for two faces. Therefore, we need to add cubes.
The final answer is
Thus, the correct answer is D.
10.
Zara has a collection of marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
Answer: C
Video solution:
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Written solution:
Let and represent the Steelie and the Tiger in some order, with always coming before
To place and we can have the following cases:
__, __ ;
__, __, ;
__, __,
leaving 3 configurations, with "__" being the reserved spots for the other marbles. With any other configuration, we have and next to each other or before
Now, each configuration can have either the Steelie be and the Tiger be or Steelie be and the Tiger be This doubles the number of configurations we have, making
Also, each configuration can have either the Bumblebee be the first spot and Aggie be the second spot or vice versa. This again doubles the number of configurations we have, making
Thus, the correct answer is C.
11.
After school, Maya and Naomi headed to the beach, miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?
Answer: E
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Written solution:
Naomi traveled miles in minutes. This ratio is
Maya traveled miles in minutes. This ratio is
This difference is
Thus, the correct answer is C.
12.
For a positive integer the factorial notation represents the product of the integers from to For example:
What value of satisfies the following equation?
Answer: A
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Written solution:
Note first that
With that in mind, further observe that:
Since we know
Using our note from above, we know that so
Thus, the correct answer is A.
13.
Jamal has a drawer containing green socks, purple socks, and orange socks. After adding more purple socks, Jamal noticed that there is now a chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
Answer: B
Video solution:
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Written solution:
Suppose Jamal adds purple socks. Then, there will be purple socks.
Also, since there are total socks to begin with, we have socks after adding the socks.
Since we have a chance of choosing a purple sock afterwards, we know
Solving for yields:
Therefore, socks are added.
Thus, the correct answer is B.
14.
There are cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all cities?
Answer: D
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Written solution:
Looking at the horizontal dashed line, the average population is around
Since there are cities, and the average population is we know:
Multiplication yields that the total population is
Thus, the correct answer is D.
15.
Suppose of equals of What percentage of is
Answer: C
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Written solution:
If a number is percent of then
This means that
Dividing each side by gives that
Therefore, we know that is of
Thus, the correct answer is C.
16.
Suppose Each of the points and in the figure below represents a different digit from to Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is What is the digit represented by B?
Answer: E
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Written solution:
Each number is added once per line it is on. Every point is on lines except for which is on
This means so
Since are unique digits from to each digit is represented exactly once, making
With we know so
Thus, the correct answer is E.
17.
How many factors of have more than factors? (As an example, has factors, namely and )
Answer: B
Video solution:
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Written solution:
Let's begin by firstly simply factoring :
These twelve factors of can be classified by the number of their factors:
- has one factor
- and has two factors
- has three (distinct) factors
Thus, all the remaining seven numbers must have more than three factors.
Thus, the correct answer is B.
18.
Rectangle is inscribed in a semicircle with diameter as shown in the figure. Let and let What is the area of
Answer: A
Video solution:
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Written solution:
Since is the diameter of the semicircle, we know the length of the diameter is and so the radius is Let be the center of the diameter.
The length from therefore is
Since is on we know
Also, since we have a semicircle, we know
Finally, since is a rectangle, we know is a right angle. This means we can find by the Pythagorean Theorem. We know
As such, the area of the rectangle is
Thus, the correct answer is A.
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
Answer: B
Video solution:
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Written solution:
For a number to be divisible by the number must be divisible by and by
First, to ensure the number is divisible by it must end in or in
Since the digits alternate between two distinct digits, we call the other digit
This would make our number either or
We can ignore the numbers in the form as they would reduce to a digit number.
As such, we know our number is for some digit
To ensure our number is also a multiple of the sum of the digits must be a multiple of
The sum of our digits is Since is a multiple of all that is required is that is a multiple of
This means can be or
Therefore, we have solutions.
Thus, the correct answer is B.
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?
\renewcommand{\arraystretch}{1.5} \begin{array}{|c|c|} \hline \text{Tree 1} & \text{__ meters} \\ \text{Tree 2} & 11 \text{ meters} \\ \text{Tree 3} & \text{__ meters} \\ \text{ Tree 4 }& \text{__ meters} \\ \text{Tree 5} & \text{__ meters} \\ \hline \text{Average height} & \text{__.2 meters} \\ \hline \end{array}
Answer: B
Video solution:
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Written solution:
We know that tree 2 is meters tall, and since the trees on either side of any given true were said to be either double the height or half the height, we can conclude that tree and three must both be meters tall, as if they were half that of tree they would not be integers.
This gives us only four possibilities for the remaining two trees, of which we simply test each case for an average that ends in "":
- Tree is meters; Tree is meters:
- Tree is meters; Tree is meters:
- Tree is meters; Tree is meters:
- Tree is meters; Tree is meters:
This means that Tree is not an integer, as therefore this case is invalid and discarded.
The only one of these cases that has an average that ends in "" is the case. As such, the average height is meters.
Thus, the correct answer is B.
21.
A game board consists of squares that alternate between blank and colored. 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 blank squares in the row above. How many -step paths are there from to (The figure shows a sample path.)
Answer: A
Video solution:
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Written solution:
Every move must go either down or up and either left or right. Since the final position is moves up and we move times, every move must go up. Since after moves, we are on the th row, we can use the th row to construct the ways to get to each position from the previous moves. Each move can come from the down-left or down-right direction, so we get the ways to get to a point by adding the number of paths from those two directions.
We have then constructed the ways to get to each point from assuming we always go up. The circled point is so we have ways to get to
Thus, the correct answer is A.
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 \begin{gather*} N \to \text{ __ } \to \text{ __ } \to \text{ __ } \\ \to \text{ __ } \to \text{ __ } \to 1 \end{gather*}
Answer: E
Video solution:
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Written solution:
To see which numbers we can make by inverting it, let's make an inverting machine.
This would take and yield either if is even (which it always is) or if is an odd integer. Note that is an integer only if Also, if is even, then is odd. That would mean would be odd. Therefore, our inverter machine yields and also if and is even.
Now, we must see what the inverting machine can yield after moves:
We can only get
From we can only get
From we can get and
From we can get only ; from we can only get
From we can get only ; from we can get and
From we can get and ; we can get ; from we can get
Move can yield and and their sum is
Thus, the correct answer is E.
23.
Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?
Answer: B
Video solution:
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Written solution:
First, we can calculate the number of way to just give out awards, without making sure every student has at least This is
Next we subtract the number of distributions without everyone having at least
This can be counted as the number of distributions with at most people having every award.
We can sort this by two cases. Case 1 has exactly 2 people having at least one award. Case 2 has exactly 1 person having at least one award.
Case 1: If two people have all the awards, there are ways to distribute the awards amongst these two people. However, we must subtract as that has only one person owning all the awards, which is outside the definiton of this case. This has combinations. Finally, we multiply this by as there are ways to choose the group of two people. This case yields distributions
Case 2: If one people has all the awards, there is way to distribute the awards amongst that person. Finally, we multiply this by as there are ways to choose the person. This case yields distributions.
There are a total of cases to remove. Therefore, the answer is
Thus, the correct answer is B.
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
Answer: A
Video solution:
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Written solution:
First, since there are squares.
Since we aren't given an area of the large square and only the ratios within it, we can define the area to be Since the total area of the gray is the area for each square is
Therefore, the side length of each tile is:
Now, there are tiles and borders, so,
Since we have
Finally,
Thus, the correct answer is A.
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?
Answer: A
Video solution:
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Written solution:
We represent the lengths of each square as and respectively. The length of the rectangle is as these squares span the entirety of a side of the large rectangle. Therefore,
Also, the height of the large rectangle is the sum of the height of and Now, note that the sum of height of and is so height of is equal to Therefore, the height of the large rectangle is which means Subtracting both of our results yields This would mean
Thus, the correct answer is A.