2023 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.
What is the value of
Answer: D
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Written solution:
We can simplify this as follows.
Thus, D is the correct answer.
2.
A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?
Answer: E
Video solution:
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Written solution:
We can unfold the cut up paper to achieve the following figure.
Thus, E is the correct answer.
3.
Wind chill is a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be found using this calculation: Where represents the wind chill, respresents air temperature measured in degrees Fahrenheit and represents wind speed measured in miles per hour (mph).
Suppose the air temperature is and the wind speed is mph. Which of the following is closest to the approximate wind chill?
Answer: B
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Written solution:
Using the formula, we can calculate the wind chill to be
Thus, B is the correct answer.
4.
The numbers from to are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number How many of these four numbers are prime?
Answer: D
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Written solution:
We can fill in the other numbers to get the complete grid.
From this, we can see that the only prime numbers in the shaded boxes are and
Thus, D is the correct answer.
5.
A lake contains trout, along with a variety of other fish. When a marine biologist catches and releases a sample of fish from the lake, are identified as trout. Assume that the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake?
Answer: B
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Written solution:
Note that This means that a sixth of the fish in the lake are trout, or in other words, the total number of fish is times the number of trout.
Therefore, there are fish in the lake.
Thus, B is the correct answer.
6.
The digits and are placed in the expression below, one digit per box. What is the maximum possible value of the expression?
Answer: C
Video solution:
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Written solution:
Note that we do not want as a base, since that would make the expression equal to
This means that must be an exponent. The number whose exponent is will automatically evaluate to
Therefore, we should minimize this number, so we can put
Now the other term is or Clearly, is larger, so we should use that.
This gives us a final value of
Thus, C is the correct answer.
7.
A rectangle, with sides parallel to the -axis and -axis, has opposite vertices located at and A line is drawn through points and Another line is drawn through points and How many points on the rectangle lie on at least one of the two lines?
Answer: B
Video solution:
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Written solution:
We can graph the two lines.
From this, we see that only the top left corner of the rectangle intersects either line.
Thus, B is the correct answer.
8.
Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against each of the other three players exactly twice. Shown below are the win-loss records for the players. The numbers and represent a win or loss, respectively. For example, Lola won five matches and lost the fourth match. What was Tiyo's win-loss record?
Answer: A
Video solution:
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Written solution:
Note that for each round, there are going to winners and losers (two matches each with a winner and loser).
This means that for each match, there should be ones and zeros.
Using this fact, we can deduce that Tiyo's win-loss record is
Thus, A is the correct answer.
9.
Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between and meters?
Answer: B
Video solution:
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Written solution:
The first time that she hits an elevation of meters is at seconds.
She then dips below meters after seconds. This adds seconds to the total answer.
Malaika then goes above meters at seconds. She hits meters again at seconds.
This adds more seconds to the total. She finally dips below meters for the last time at seconds.
She then falls below meters at seconds, finally adding seconds to the total time.
The desired answer is therefore Thus, B is the correct answer.
10.
Harold made a plum pie to take on a picnic. He was able to eat only of the pie, and he left the rest for his friends. A moose came by and of what Harold left behind. After that, a porcupine ate of what the moose left behind. How much of the original pie still remained after the porcupine left?
Answer: D
Video solution:
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Written solution:
Harold left of the pie for his friends.
The moose ate of the pie, leaving of the pie.
Finally, the porcupine ate of the pie. This leaves of the pie.
Thus, D is the correct answer.
11.
NASA's Perseverance Rover was launched on July After traveling miles, it landed on Mars in Jezero Crater about months later. Which of the following is closest to the Rover's average interplanetary speed in miles per hour?
Answer: C
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Written solution:
We can round the distance up to miles. We can also approximate months as days. This means that the Rover traveled miles per day. Dividing this by to get the speed in miles per hour yields which is close to
Thus, C is the correct answer.
12.
The figure below shows a large unshaded circle with a number of smaller unshaded and shaded circles in its interior. What fraction of the interior of the large unshaded circle is shaded?
Answer: B
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Written solution:
WLOG, assume that each square has a side length of units.
This means that there are shaded unit circles, which total to area.
There is also a shaded circle with radius with two white circles of radius inside.
This gives us an extra shaded area of
Adding these values together yields The area of the large white circle is Therefore, the desired fraction is
Thus, B is the correct answer.
13.
Along the route of a bicycle race, water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also repair stations evenly spaced between the start and finish lines. The rd water station is located miles after the st repair station. How long is the race in miles?
Answer: D
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Written solution:
The rd water station is located of the way along the race (the water stations split the race up into equal spaces).
The first repair station is located of the way along the race. The distance between the two is the length of the race. We know that this equals which means the race is miles long.
Thus, D is the correct answer.
14.
Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of -cent, -cent, and -cent stamps, with exactly of each type. What is the greatest number of stamps Nicolas can use to make exactly $7.10 in postage?
(Note: The amount $7.10 corresponds to dollars and cents. One dollar is worth cents.)
Answer: E
Video solution:
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Written solution:
Note that we want to get cents. Let us try to see if it is possible to use up all of the -cent and -cent stamps.
All of these two types of stamps combined would be worth cents. We then would need cents, which cannot be created with just -cent stamps.
We can, however, make cents with -cent stamps. Using only each of and -cent stamps would total cents. This means we would then need cents. This can be achieved with -cent stamps.
This lets us use stamps.
Thus, E is the correct answer.
15.
Visvam walks half a mile to get to school each day. His route consists of city blocks of equal length and he takes one minute to walk each block. Today, after walking blocks, Visvam discovers that he has to make a detour, walking blocks of equal length instead of block to reach the next corner. From the time he starts his detour, at what speed, in miles per hour, must Visvam walk in order to arrive at school at his usual time?
Answer: B
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Written solution:
If half a mile is the same as blocks, then one block is miles.
Starting from the detour, Visvam has to walk miles.
Normally, from this spot Visvam would take minutes to walk to school. Now he has to travel miles in minutes.
Note that minutes is miles. This means his speed must be mph.
Thus, B is the correct answer.
16.
The letters P, Q, and R are entered into a table according to the pattern shown below. How many Ps, Qs, and Rs will appear in the completed table.
Ps, Qs, Rs
Ps, Qs, Rs
Ps, Qs, Rs
Ps, Qs, Rs
Ps, Qs, Rs
Answer: C
Video solution:
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Written solution:
Since the bottom letters in each column will occur one more time than the third letter.
This means that the third letter in each column will occur times, whereas the other will appear times.
Using the same analysis, we can see that and appear in the third row times, whereas only appears times.
Therefore, in columns, and appear times, for a total of times.
They also appear times in columns, adding appearances to the total. This gives us a total of appearances for and
will therefore appear times.
Thus, C is the correct answer.
17.
A regular octahedron has eight equilateral triangle face with four faces meeting at each vertex. Jun will make the regular octahedron shown in the figure by folding the piece of paper below. Which number face will end up to the right of the shaded region
Answer: A
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Written solution:
Begin by observing that when folded, the sides labelled and form the bottom half of the octahedron. As such, the remaining four faces must make up the top half of the octahedron.
From here, we have narrowed down our possibilities to and We can see that will be the number to the left of the shaded region This also gives us that is to the left of
Therefore, we know that the only remaining face, must be to the right of the shaded region
Thus, A is the correct answer.
18.
Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump pads to the right or pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located pads to the right of her starting position?
Answer: D
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Written solution:
An optimal strategy would be jumping as close as possible with the right jumps and then fine tuning with the left jumps.
It will take at least jumps to get to Clearly, we cannot get to in one more jump, so A cannot be right.
With jumps, the only way to move forward is with jumps right and the one jump left, but that puts us at
This shows that B cannot happen.
Using jumps, we can jump right twice and jump left thrice, but that puts us at Jumping right rice and left twice would put us at
This shows that C cannot happen either.
Finally, with jumps, we can jump right thrice and left four times, putting us at
Thus, D is the correct answer.
19.
An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle.
Answer: C
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Written solution:
Since the inner triangle's side length is the side length of the outer triangle, its area is the area of the outer triangle.
This means that the three trapezoids are the area of the outer triangle.
Therefore, one trapezoid is the area of the outer triangle.
This makes the ratio of the areas of one trapezoid and the inner triangle
Thus, C is the correct answer.
20.
Two integers are inserted into the list to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?
Answer: D
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Written solution:
Since the median remains unchanged. One number less than and one number greater than is added.
The inequalities are strict since the mode doesn't change, which means that only can appear twice.
The current range is The new range is therefore
To maximize the smaller number, we can set it equal to The larger number is forced to be
Their sum is
Thus, D is the correct answer.
21.
Alina writes the numbers on separate cards, one number per card. She wishes to divide the cards into groups of cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?
Answer: C
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Written solution:
The sum of all the numbers is This means that the sum of each group is
Consider the group with in it. The other two numbers must add to Therefore, the other cards in this group are and or and
Case One group is
Consider the group with in it. The other numbers must add to The only option is and with the remaining cards.
The other group is then and This adds to so this case contributes one possibility.
Case One group is
Consider the group with in it. As above, the other numbers have to add to The only option is and
The final group is and which adds to This is another configuration.
We have gone through all the cases, which revealed that there are only possible groupings.
Thus, C is the correct answer.
22.
In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term in the sequence is What is the first term?
Answer: D
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Written solution:
Let be the first term and be the second.
Then we get the following sequence. From this, we get that
Factoring we get We need and to be integers. The only fifth powers that divides are and
If then which doesn't work since is not a perfect cube. Therefore,
This forces to equal
Thus, D is the correct answer.
23.
Each square in a grid is randomly filled with one of the gray-and-white tiles shown below on the right.
What is the probability that the tiling will contain a large gray diamond in one of the smaller grids? Below is an example of such a tiling.
Answer: C
Video solution:
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Written solution:
There are a total of configurations. The square that contains the large gray diamond can occur in spots (as there are four squares in the big square).
The four tiles in the square have a fixed tile. The other squares can be any tiles, which allows for configurations for the other tiles.
The total number of configurations that include the large gray diamond is therefore
The desired probability is then
Thus, C is the correct answer.
24.
Isosceles triangle has equal side lengths and In the figures below, segments are drawn parallel to so that the shaded portions of have the same area. The heights of the two unshaded portions are and units, respectively. What is the height of
Answer: A
Video solution:
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Written solution:
Let be the area of
Note that the smaller triangles are similar to This means that the ratio of their areas is the ratio of their side lengths squared.
Then we get that The left side is the area of the whole triangle minus the area of the unshaded region of the left triangle.
The right hand side is the area of the shaded triangle. We get this by finding the ratio of their side lengths (which is the same as the ratio of their heights) and squaring it.
Simplifying yields This simplifies to
Thus, A is the correct answer.
25.
Fifteen integers are arranged in order on a number line. The integers are equally spaced and have the property that and
What is the sum of the digits of
Answer: A
Video solution:
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Written solution:
Let be the common difference. If we let and we see that Since all the numbers are integers, must be at least
Also, if and we get that Once again since all the numbers are integers, is at most This tells us that is
Note that This means that must be at least for to be within the desired range.
If is greater than however, becomes greater than which is not allowed.
Now we know that and This tell us that
Therefore, sum of the digits is
Thus, A is the correct answer.